Displaying 1-10 of 11 results found.
Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.
+20
10
1, -1, 1, 0, -3, 1, 0, 3, -6, 1, 0, 0, 15, -10, 1, 0, 0, -15, 45, -15, 1, 0, 0, 0, -105, 105, -21, 1, 0, 0, 0, 105, -420, 210, -28, 1, 0, 0, 0, 0, 945, -1260, 378, -36, 1, 0, 0, 0, 0, -945, 4725, -3150, 630, -45, 1, 0, 0, 0, 0, 0, -10395, 17325, -6930, 990, -55, 1, 0, 0, 0, 0, 0, 10395, -62370, 51975, -13860, 1485, -66, 1
COMMENTS
Also the Bell transform of (-1)^n if n<2 else 0 and the inverse Bell transform of A001147(n) (adding 1,0,0,... as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016
FORMULA
E.g.f. : (1 - t)*exp(x*(t - t^2/2)) = 1 + (-1 + x)*t + (-3*x + x^2)*t^2/2! + ... - Peter Bala, Apr 08 2013
EXAMPLE
Rows begin:
1;
-1, 1;
0, -3, 1;
0, 3, -6, 1;
0, 0, 15, -10, 1;
0, 0, -15, 45, -15, 1;
0, 0, 0, -105, 105, -21, 1;
0, 0, 0, 105, -420, 210, -28, 1;
0, 0, 0, 0, 945, -1260, 378, -36, 1;
0, 0, 0, 0, -945, 4725, -3150, 630, -45, 1; ...
The columns being equal in absolute value to the rows of the matrix inverse A001497: 1;
1, 1;
3, 3, 1;
15, 15, 6, 1;
105, 105, 45, 10, 1;
945, 945, 420, 105, 15, 1; ...
MATHEMATICA
With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - t)*Exp[x*(t - t^2/2)], {t, 0, nmax}, {x, 0, nmax}], t], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 10 2018 *)
PROG
(Sage) # uses[bell_matrix from A264428] # Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(lambda n: (-1)^n if n<2 else 0, 9) # Peter Luschny, Jan 19 2016
CROSSREFS
Row sums are found in A001464 (offset 1).
Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.
(Formerly M1221 N0469)
+10
464
1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, 211799312, 997313824, 4809701440, 23758664096, 119952692896, 618884638912, 3257843882624, 17492190577600, 95680443760576, 532985208200576, 3020676745975552
COMMENTS
a(n) is also the number of n X n symmetric permutation matrices.
a(n) is also the number of matchings (Hosoya index) in the complete graph K(n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001
a(n) is also the number of independent vertex sets and vertex covers in the n-triangular graph. - Eric W. Weisstein, May 22 2017 Equivalently, this is the number of graphs on n labeled nodes with degrees at most 1. - Don Knuth, Mar 31 2008 a(n) is also the sum of the degrees of the irreducible representations of the symmetric group S_n. - Avi Peretz (njk(AT)netvision.net.il), Apr 01 2001
a(n) is the number of partitions of a set of n distinguishable elements into sets of size 1 and 2. - Karol A. Penson, Apr 22 2003 Number of tableaux on the edges of the star graph of order n, S_n (sometimes T_n). - Roberto E. Martinez II, Jan 09 2002 The Hankel transform of this sequence is A000178 (superfactorials). Sequence is also binomial transform of the sequence 1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, ... ( A001147 with interpolated zeros). - Philippe Deléham, Jun 10 2005 Row sums of the exponential Riordan array (e^(x^2/2),x). - Paul Barry, Jan 12 2006 a(n) is the number of nonnegative lattice paths of upsteps U = (1,1) and downsteps D = (1,-1) that start at the origin and end on the vertical line x = n in which each downstep (if any) is marked with an integer between 1 and the height of its initial vertex above the x-axis. For example, with the required integer immediately preceding each downstep, a(3) = 4 counts UUU, UU1D, UU2D, U1DU. - David Callan, Mar 07 2006 Proof of the recurrence relation a(n) = a(n-1) + (n-1)*a(n-2): number of involutions of [n] containing n as a fixed point is a(n-1); number of involutions of [n] containing n in some cycle (j, n), where 1 <= j <= n-1, is (n-1) times the number of involutions of [n] containing the cycle (n-1 n) = (n-1)*a(n-2). - Emeric Deutsch, Jun 08 2009 Number of ballot sequences (or lattice permutations) of length n. A ballot sequence B is a string such that, for all prefixes P of B, h(i) >= h(j) for i < j, where h(x) is the number of times x appears in P. For example, the ballot sequences of length 4 are 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1231, and 1234. The string 1221 does not appear in the list because in the 3-prefix 122 there are two 2's but only one 1. (Cf. p. 176 of Bruce E. Sagan: "The Symmetric Group"). - Joerg Arndt, Jun 28 2009 Number of standard Young tableaux of size n; the ballot sequences are obtained as a length-n vector v where v_k is the (number of the) row in which the number r occurs in the tableaux. - Joerg Arndt, Jul 29 2012 Number of factorial numbers of length n-1 with no adjacent nonzero digits. For example the 10 such numbers (in rising factorial radix) of length 3 are 000, 001, 002, 003, 010, 020, 100, 101, 102, and 103. - Joerg Arndt, Nov 11 2012 Also called restricted Stirling numbers of the second kind (see Mezo). - N. J. A. Sloane, Nov 27 2013 a(n) is the number of permutations of [n] that avoid the consecutive patterns 123 and 132. Proof. Write a self-inverse permutation in standard cycle form: smallest entry in each cycle in first position, first entries decreasing. For example, (6,7)(3,4)(2)(1,5) is in standard cycle form. Then erase parentheses. This is a bijection to the permutations that avoid consecutive 123 and 132 patterns. - David Callan, Aug 27 2014 Getu (1991) says these numbers are also known as "telephone numbers". - N. J. A. Sloane, Nov 23 2015 a(n) is the number of elements x in S_n such that x^2 = e where e is the identity. - Jianing Song, Aug 22 2018 a(n) is the number of congruence orbits of upper-triangular n X n matrices on skew-symmetric matrices, or the number of Borel orbits in largest sect of the type DIII symmetric space SO_{2n}(C)/GL_n(C). Involutions can also be thought of as fixed-point-free partial involutions. See [Bingham and Ugurlu] link. - Aram Bingham, Feb 08 2020 Apparently a(n) = b*c where b is odd iff a(n+b) (when a(n) is defined) is divisible by b.
Apparently a(n) = 2^(f(n mod 4)+floor(n/4))*q where f:{0,1,2,3}->{0,1,2} is given by f(0),f(1)=0, f(2)=1 and f(3)=2 and q is odd. (End)
a(n) is the n-th initial moment of the normal distribution with mean 1 and variance 1. This follows because the moment generating function of that distribution is the e.g.f. of the sequence of the a(n)'s.
The recurrence a(n) = a(n-1) + (n-1)*a(n-2) also follows, by writing E(Z+1)^n=EZ(Z+1)^(n-1)+E(Z+1)^(n-1), where Z is a standard normal random variable, and then taking the first of the latter two integrals by parts. (End)
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 32, 911.
S. Chowla, The asymptotic behavior of solutions of difference equations, in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), Vol. I, 377, Amer. Math. Soc., Providence, RI, 1952.
W. Fulton, Young Tableaux, Cambridge, 1997.
D. E. Knuth, The Art of Computer Programming, Vol. 3, Section 5.1.4, p. 65.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
T. Muir, A Treatise on the Theory of Determinants. Dover, NY, 1960, p. 6.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 86.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
LINKS
S. Bilotta, E. Pergola, R. Pinzani, and S. Rinaldi, Recurrence Relations, Succession Rules, and the Positivity Problem, in Language and Automata Theory and Applications, 9th International Conference, LATA 2015, Nice, France, March 2-6, 2015, Proceedings, Pages 499-510, Lecture Notes Comp. Sci. Vol. 8977. Eric Weisstein's World of Mathematics, MatchingAllan Wentink, Symmetry with Independent Blocks, on the late Ralph E. Griswold webpage of sample-chapters of an unfinished book about weaving, http://www.cs.arizona.edu/patterns/weaving/webdocs.html. [Cached copy]
FORMULA
D-finite with recurrence a(0) = a(1) = 1, a(n) = a(n-1) + (n-1)*a(n-2) for n>1.
E.g.f.: exp(x+x^2/2).
a(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)!*2^k*k!).
a(m+n) = Sum_{k>=0} k!*binomial(m, k)*binomial(n, k)*a(m-k)*a(n-k). - Philippe Deléham, Mar 05 2004 The e.g.f. y(x) satisfies y^2 = y''y' - (y')^2.
a(n) ~ c*(n/e)^(n/2)exp(n^(1/2)) where c=2^(-1/2)exp(-1/4). [Chowla]
a(n) = HermiteH(n, 1/(sqrt(2)*i))/(-sqrt(2)*i)^n, where HermiteH are the Hermite polynomials. - Karol A. Penson, May 16 2002 a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2. - Paul Barry, Jan 12 2006 For asymptotics see the Robinson paper.
O.g.f.: A(x) = 1/(1-x-1*x^2/(1-x-2*x^2/(1-x-3*x^2/(1-... -x-n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006 a(n) = (n-1)*a(n-2) + a(n-1); e.g., a(7) = 232 = 6*26 + 76.
Starting with offset 1 = eigensequence of triangle A128229. (End) a(n) = (1/sqrt(2*Pi))*Integral_{x=-oo..oo} exp(-x^2/2)*(x+1)^n. - Groux Roland, Mar 14 2011 Continued fractions:
E.g.f.: 1+x*(2+x)/(2*G(0)-x*(2+x)) where G(k)=1+x*(x+2)/(2+2*(k+1)/G(k+1)).
G.f.: 1/(U(0) - x) where U(k) = 1 + x*(k+1) - x*(k+1)/(1 - x/U(k+1)).
G.f.: 1/Q(0) where Q(k) = 1 + x*k - x/(1 - x*(k+1)/Q(k+1)).
G.f.: -1/(x*Q(0)) where Q(k) = 1 - 1/x - (k+1)/Q(k+1).
G.f.: T(0)/(1-x) where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x)^2/T(k+1)). (End)
a(n) ~ sqrt(2)/2 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 07 2014 a(n) = Sum_{k=0..n} s(n,k)*(-1)^(n-k)*2^k*B(k,1/2), where the s(n,k) are (signless) Stirling numbers of the first kind, and the B(n,x) = Sum_{k=0..n} S(n,k)*x^k are the Stirling polynomials, where the S(n,k) are the Stirling numbers of the second kind. - Emanuele Munarini, May 16 2014 a(n) = hyper2F0([-n/2,(1-n)/2],[],2). - Peter Luschny, Aug 21 2014 0 = a(n)*(+a(n+1) + a(n+2) - a(n+3)) + a(n+1)*(-a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Aug 22 2014 a(n+k) == a(n) (mod k) for all n >= 0 and all positive odd integers k.
Hence for each odd k, the sequence obtained by taking a(n) modulo k is a periodic sequence and the exact period divides k. For example, taking a(n) modulo 7 gives the purely periodic sequence [1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, ...] of period 7. For similar results see A047974 and A115329. (End)
EXAMPLE
Sequence starts 1, 1, 2, 4, 10, ... because possibilities are {}, {A}, {AB, BA}, {ABC, ACB, BAC, CBA}, {ABCD, ABDC, ACBD, ADCB, BACD, BADC, CBAD, CDAB, DBCA, DCBA}. - Henry Bottomley, Jan 16 2001 G.f. = 1 + x + 2*x^2 + 4*x^4 + 10*x^5 + 26*x^6 + 76*x^7 + 232*x^8 + 764*x^9 + ...
The a(4) = 10 standard Young tableaux:
1 2 3 4
.
1 2 1 3 1 2 3 1 2 4 1 3 4
3 4 2 4 4 3 2
.
1 2 1 3 1 4
3 2 2
4 4 3
.
1
2
3
4
The a(0) = 1 through a(4) = 10 set partitions into singletons or pairs:
{} {{1}} {{1,2}} {{1},{2,3}} {{1,2},{3,4}}
{{1},{2}} {{1,2},{3}} {{1,3},{2,4}}
{{1,3},{2}} {{1,4},{2,3}}
{{1},{2},{3}} {{1},{2},{3,4}}
{{1},{2,3},{4}}
{{1,2},{3},{4}}
{{1},{2,4},{3}}
{{1,3},{2},{4}}
{{1,4},{2},{3}}
{{1},{2},{3},{4}}
(End)
MAPLE
A000085 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else procname(n-1)+(n-1)*procname(n-2); fi; end;
with(combstruct):ZL3:=[S, {S=Set(Cycle(Z, card<3))}, labeled]:seq(count(ZL3, size=n), n=0..25); # Zerinvary Lajos, Sep 24 2007 with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, m>=card))}, labeled]; end: A:=a(2):seq(count(A, size=n), n=0..25); # Zerinvary Lajos, Jun 11 2008
MATHEMATICA
<<Combinatorica`; [M[Star[n]]]
p[0, x] = 1; p[1, x] = x; p[k_, x_] := p[k, x] = x*p[k - 1, x] + (k - 1)*p[k - 2, x]; Table[Sum[CoefficientList[p[n, x], x][[m]], {m, 1, n + 1}], {n, 0, 15}] (* Roger L. Bagula, Oct 06 2006 *) With[{nn=30}, CoefficientList[Series[Exp[x+x^2/2], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, May 28 2013 *) a[ n_] := Sum[(2 k - 1)!! Binomial[ n, 2 k], {k, 0, n/2}]; (* Michael Somos, Jun 01 2013 *) a[ n_] := If[ n < 0, 0, HypergeometricU[ -n/2, 1/2, -1/2] / (-1/2)^(n/2)]; (* Michael Somos, Jun 01 2013 *) a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ x + x^2 / 2], {x, 0, n}]]; (* Michael Somos, Jun 01 2013 *) Table[(I/Sqrt[2])^n HermiteH[n, -I/Sqrt[2]], {n, 0, 100}] (* Emanuele Munarini, Mar 02 2016 *) a[n_] := Sum[StirlingS1[n, k]*2^k*BellB[k, 1/2], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 18 2017, after Emanuele Munarini *) RecurrenceTable[{a[n] == a[n-1] + (n-1)*a[n-2], a[0] == 1, a[1] == 1}, a, {n, 0, 20}] (* Joan Ludevid, Jun 17 2022 *) sds[{}]:={{}}; sds[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sds[Complement[set, s]]]/@Cases[Subsets[set, {1, 2}], {i, ___}]; Table[Length[sds[Range[n]]], {n, 0, 10}] (* Gus Wiseman, Jan 11 2021 *)
PROG
(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( x + x^2 / 2 + x * O(x^n)), n))}; /* Michael Somos, Nov 15 2002 */ (PARI) N=66; x='x+O('x^N); egf=exp(x+x^2/2); Vec(serlaplace(egf)) \\ Joerg Arndt, Mar 07 2013 (Haskell) a000085 n = a000085_list !! n
a000085_list = 1 : 1 : zipWith (+)
(Maxima) B(n, x):=sum(stirling2(n, k)*x^k, k, 0, n);
a(n):=sum(stirling1(n, k)*2^k*B(k, 1/2), k, 0, n);
(Maxima) makelist((%i/sqrt(2))^n*hermite(n, -%i/sqrt(2)), n, 0, 12); /* Emanuele Munarini, Mar 02 2016 */ (Sage) A000085 = lambda n: hypergeometric([-n/2, (1-n)/2], [], 2) (Sage) def a85(n): return sum(factorial(n) / (factorial(n-2*k) * 2**k * factorial(k)) for k in range(1+n//2))
(Python)
from math import factorial
def A000085(n): return sum(factorial(n)//(factorial(n-(k<<1))*factorial(k)*(1<<k)) for k in range((n>>1)+1)) # Chai Wah Wu, Aug 31 2023
CROSSREFS
See also A005425 for another version of the switchboard problem. A320663/ A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Triangle read by rows: coefficients of modified Hermite polynomials.
+10
24
1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1, 0, 945, 0, 1260, 0, 378, 0, 36, 0, 1, 945, 0, 4725, 0, 3150, 0, 630, 0, 45, 0, 1, 0, 10395, 0, 17325, 0, 6930, 0, 990, 0, 55, 0, 1
COMMENTS
T(n,k) is the number of involutions of {1,2,...,n}, having k fixed points (0 <= k <= n). Example: T(4,2)=6 because we have 1243,1432,1324,4231,3214 and 2134. - Emeric Deutsch, Oct 14 2006 Riordan array [exp(x^2/2),x]. - Paul Barry, Nov 06 2008 Same as triangle of Bessel numbers of second kind, B(n,k) (see Cheon et al., 2013). - N. J. A. Sloane, Sep 03 2013 The modified Hermite polynomial h(n,x) (as in the Formula section) is the numerator of the rational function given by f(n,x) = x + (n-2)/f(n-1,x), where f(x,0) = 1. - Clark Kimberling, Oct 20 2014 Second lower diagonal T(n,n-2) equals positive triangular numbers A000217 \ {0}. - M. F. Hasler, Oct 23 2014 T(n,k) is the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the Brauer monoid of degree n.
For n < k with n == k (mod 2), T(n,k) is the rank (minimal size of a generating set) and idempotent rank (minimal size of an idempotent generating set) of the ideal consisting of all rank <= k elements of the Brauer monoid. (End)
This array provides the coefficients of a Laplace-dual sequence H(n,x) of the Dirac delta function, delta(x), and its derivatives, formed by taking the inverse Laplace transform of these modified Hermite polynomials. H(n,x) = h(n,D) delta(x) with h(n,x) as in the examples and the lowering and raising operators L = -x and R = -x + D = -x + d/dx such that L H(n,x) = n * H(n-1,x) and R H(n,x) = H(n+1,x). The e.g.f. is exp[t H(.,x)] = e^(t^2/2) e^(t D) delta(x) = e^(t^2/2) delta(x+t). - Tom Copeland, Oct 02 2016 This triangle is the reverse of that in Table 2 on p. 7 of the Artioli et al. paper and Table 6.2 on p. 234 of Licciardi's thesis, with associations to the telephone numbers. - Tom Copeland, Jun 18 2018 and Jul 08 2018 See A344678 for connections to a Heisenberg-Weyl algebra of differential operators, matching and independent edge sets of the regular n-simplices with partially labeled vertices, and telephone switchboard scenarios. - Tom Copeland, Jun 02 2021
FORMULA
h(k, x) = (-I/sqrt(2))^k * H(k, I*x/sqrt(2)), H(n, x) the Hermite polynomials ( A060821, A059343). T(n,k) = n!/(2^((n-k)/2)*((n-k)/2)!k!) if n-k >= 0 is even; 0 otherwise. - Emeric Deutsch, Oct 14 2006 G.f.: 1/(1-x*y-x^2/(1-x*y-2*x^2/(1-x*y-3*x^2/(1-x*y-4*x^2/(1-... (continued fraction). - Paul Barry, Apr 10 2009 Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n >= 1 T(n,k) = T(n-1,k-1) + (k+1)*T(n-1,k+1). - Peter Luschny, Oct 06 2012 The row polynomials P(n,x) = (a. + x)^n, umbrally evaluated with (a.)^n = a_n = aerated A001147, are an Appell sequence with dP(n,x)/dx = n * P(n-1,x). The umbral compositional inverses (cf. A001147) of these polynomials are given by the same polynomials signed, A066325. - Tom Copeland, Nov 15 2014 The odd rows are (2x^2)^n x n! L(n,-1/(2x^2),1/2), and the even, (2x^2)^n n! L(n,-1/(2x^2),-1/2) in sequence with n= 0,1,2,... and L(n,x,a) = Sum_{k=0..n} binomial(n+a,k+a) (-x)^k/k!, the associated Laguerre polynomial of order a. The odd rows are related to A130757, and the even to A176230 and A176231. Other versions of this entry are A122848, A049403, A096713 and A104556, and reversed A100861, A144299, A111924. With each non-vanishing diagonal divided by its initial element A001147(n), this array becomes reversed, aerated A034839. Create four shift and stretch matrices S1,S2,S3, and S4 with all elements zero except S1(2n,n) = 1 for n >= 1, S2(n,2n) = 1 for n >= 0, S3(2n+1,n) = 1 for n >= 1, and S4(n,2n+1) = 1 for n >= 0. Then this entry's lower triangular matrix is T = Id + S1 * ( A176230-Id) * S2 + S3 * (unsigned A130757-Id) * S4 with Id the identity matrix. The sandwiched matrices have infinitesimal generators with the nonvanishing subdiagonals A000384(n>0) and A014105(n>0). As an Appell sequence, the lowering and raising operators are L = D and R = x + dlog(exp(D^2/2))/dD = x + D, where D = d/dx, L h(n,x) = n h(n-1,x), and R h(n,x) = h(n+1,x), so R^n 1 = h(n,x). The fundamental moment sequence has the e.g.f. e^(t^2/2) with coefficients a(n) = aerated A001147, i.e., h(n,x) = (a. + x)^n, as noted above. The raising operator R as a matrix acting on o.g.f.s (formal power series) is the transpose of the production matrix P below, i.e., (1,x,x^2,...)(P^T)^n (1,0,0,...)^T = h(n,x). For characterization as a Riordan array and associations to combinatorial structures, see the Barry link and the Yang and Qiao reference. For relations to projective modules, see the Sazdanovic link.
(End)
From the Appell formalism, e^(D^2/2) x^n = h_n(x), the n-th row polynomial listed below, and e^(-D^2/2) x^n = u_n(x), the n-th row polynomial of A066325. Then R = e^(D^2/2) * x * e^(-D^2/2) is another representation of the raising operator, implied by the umbral compositional inverse relation h_n(u.(x)) = x^n. - Tom Copeland, Oct 02 2016 h_n(x) = p_n(x-1), where p_n(x) are the polynomials of A111062, related to the telephone numbers A000085. - Tom Copeland, Jun 26 2018 In the power basis x^n, the matrix infinitesimal generator M = A132440^2/2, when acting on a row vector for an o.g.f., is the matrix representation for the differential operator D^2/2. e^{M} gives the coefficients of the Hermite polynomials of this entry.
The only nonvanishing subdiagonal of M, the second subdiagonal (1,3,6,10,...), gives, aside from the initial 0, the triangular numbers A000217, the number of edges of the n-dimensional simplices with (n+1) vertices. The perfect matchings of these simplices are the aerated odd double factorials A001147 noted above, the moments for the Hermite polynomials. The polynomials are also generated from A036040 with x[1] = x, x[2] = 1, and the other indeterminates equal to zero. (End)
EXAMPLE
h(0,x) = 1
h(1,x) = x
h(2,x) = x^2 + 1
h(3,x) = x^3 + 3*x
h(4,x) = x^4 + 6*x^2 + 3
h(5,x) = x^5 + 10*x^3 + 15*x
h(6,x) = x^6 + 15*x^4 + 45*x^2 + 15
Triangle begins
1,
0, 1,
1, 0, 1,
0, 3, 0, 1,
3, 0, 6, 0, 1,
0, 15, 0, 10, 0, 1,
15, 0, 45, 0, 15, 0, 1
Production array starts
0, 1,
1, 0, 1,
0, 2, 0, 1,
0, 0, 3, 0, 1,
0, 0, 0, 4, 0, 1,
0, 0, 0, 0, 5, 0, 1 (End)
MAPLE
T:=proc(n, k) if n-k mod 2 = 0 then n!/2^((n-k)/2)/((n-k)/2)!/k! else 0 fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Oct 14 2006
MATHEMATICA
nn=10; a=y x+x^2/2!; Range[0, nn]!CoefficientList[Series[Exp[a], {x, 0, nn}], {x, y}]//Grid (* Geoffrey Critzer, May 08 2012 *) H[0, x_] = 1; H[1, x_] := x; H[n_, x_] := H[n, x] = x*H[n-1, x]-(n-1)* H[n-2, x]; Table[CoefficientList[H[n, x], x], {n, 0, 11}] // Flatten // Abs (* Jean-François Alcover, May 23 2016 *) T[ n_, k_] := If[ n < 0, 0, Coefficient[HermiteH[n, x I/Sqrt[2]] (Sqrt[1/2]/I)^n, x, k]]; (* Michael Somos, May 10 2019 *)
PROG
(Sage)
M = matrix(ZZ, dim, dim)
for n in (0..dim-1): M[n, n] = 1
for n in (1..dim-1):
for k in (0..n-1):
M[n, k] = M[n-1, k-1]+(k+1)*M[n-1, k+1]
return M
(PARI) T(n, k)=if(k<=n && k==Mod(n, 2), n!/k!/(k=(n-k)/2)!>>k) \\ M. F. Hasler, Oct 23 2014 (Python)
import sympy
from sympy import Poly
from sympy.abc import x, y
def H(n, x): return 1 if n==0 else x if n==1 else x*H(n - 1, x) - (n - 1)*H(n - 2, x)
def a(n): return [abs(cf) for cf in Poly(H(n, x), x).all_coeffs()[::-1]]
(Python)
def Trow(n: int) -> list[int]:
row: list[int] = [0] * (n + 1); row[n] = 1
for k in range(n - 2, -1, -2):
row[k] = (row[k + 2] * (k + 2) * (k + 1)) // (n - k)
CROSSREFS
Row sums (polynomial values at x=1) are A000085. Cf. A000384, A014105, A034839, A049403, A096713, A100861, A104556, A122848, A130757, A176230, A176231.
Exponential Riordan array (1, x(1+x/2)).
+10
23
1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 0, 15, 10, 1, 0, 0, 0, 15, 45, 15, 1, 0, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0
COMMENTS
T(n,k) is the number of self-inverse permutations of {1,2,...,n} having exactly k cycles. - Geoffrey Critzer, May 08 2012 Also the inverse Bell transform of the double factorial of odd numbers Product_{k= 0..n-1} (2*k+1) ( A001147). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015
FORMULA
Number triangle T(n,k) = k!*C(n,k)/((2k-n)!*2^(n-k)).
Triangle equals the matrix product A008275* A039755. Equivalently, the n-th row polynomial R(n,x) is given by the Type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} P(n,2*k+1)*(x/2)^k/k!, where P(n,x) = x*(x-1)*...*(x-n+1) denotes the falling factorial polynomial. Cf. A113278. - Peter Bala, Jun 23 2014 E.g.f. for the m-th column: (x^2/2+x)^m/m!.
T(n,k) = T(n-1,k-1) + (n-1)*T(n-2,k-1) for n>1 and k=1..n, T(0,0) = 1. (End)
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 0 3 1
0 0 3 6 1
0 0 0 15 10 1
0 0 0 15 45 15 1
0 0 0 0 105 105 21 1
0 0 0 0 105 420 210 28 1
0 0 0 0 0 945 1260 378 36 1
As noted above, a(n) is the number of set partitions of {1..n} into k singletons or pairs. This is also the number of set partitions of subsets of {1..n} into n - k pairs. In the first case, row n = 5 counts the following set partitions:
{{1},{2,3},{4,5}} {{1},{2},{3},{4,5}} {{1},{2},{3},{4},{5}}
{{1,2},{3},{4,5}} {{1},{2},{3,4},{5}}
{{1,2},{3,4},{5}} {{1},{2,3},{4},{5}}
{{1,2},{3,5},{4}} {{1,2},{3},{4},{5}}
{{1},{2,4},{3,5}} {{1},{2},{3,5},{4}}
{{1},{2,5},{3,4}} {{1},{2,4},{3},{5}}
{{1,3},{2},{4,5}} {{1},{2,5},{3},{4}}
{{1,3},{2,4},{5}} {{1,3},{2},{4},{5}}
{{1,3},{2,5},{4}} {{1,4},{2},{3},{5}}
{{1,4},{2},{3,5}} {{1,5},{2},{3},{4}}
{{1,4},{2,3},{5}}
{{1,4},{2,5},{3}}
{{1,5},{2},{3,4}}
{{1,5},{2,3},{4}}
{{1,5},{2,4},{3}}
In the second case, we have:
{{1,2},{3,4}} {{1,2}} {}
{{1,2},{3,5}} {{1,3}}
{{1,2},{4,5}} {{1,4}}
{{1,3},{2,4}} {{1,5}}
{{1,3},{2,5}} {{2,3}}
{{1,3},{4,5}} {{2,4}}
{{1,4},{2,3}} {{2,5}}
{{1,4},{2,5}} {{3,4}}
{{1,4},{3,5}} {{3,5}}
{{1,5},{2,3}} {{4,5}}
{{1,5},{2,4}}
{{1,5},{3,4}}
{{2,3},{4,5}}
{{2,4},{3,5}}
{{2,5},{3,4}}
(End)
MAPLE
# The function BellMatrix is defined in A264428. BellMatrix(n -> `if`(n<2, 1, 0), 9); # Peter Luschny, Jan 27 2016
MATHEMATICA
t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten
(* Second program: *)
rows = 12;
t = Join[{1, 1}, Table[0, rows]];
T[n_, k_] := BellY[n, k, t];
sbs[{}]:={{}}; sbs[set:{i_, ___}]:=Join@@Function[s, (Prepend[#1, s]&)/@sbs[Complement[set, s]]]/@Cases[Subsets[set], {i}|{i, _}];
Table[Length[Select[sbs[Range[n]], Length[#]==k&]], {n, 0, 6}, {k, 0, n}] (* Gus Wiseman, Jan 12 2021 *)
PROG
(PARI) {T(n, k)=if(2*k<n||k>n, 0, n!/(2*k-n)!/(n-k)!*2^(k-n))} /* Michael Somos, Oct 03 2006 */ (Sage) # uses[inverse_bell_transform from A265605] multifact_2_1 = lambda n: prod(2*k + 1 for k in (0..n-1))
inverse_bell_matrix(multifact_2_1, 9) # Peter Luschny, Dec 31 2015
CROSSREFS
Same as A049403 but with a first column k = 0. The same set partitions counted by number of pairs are A100861. Reversing rows gives A111924 (without column k = 0). A047884 counts standard Young tableaux by size and greatest row length.
A238123 counts standard Young tableaux by size and least row length.
A320663/ A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Triangular table of coefficients of Laguerre-Sonin polynomials n!*2^n*Lag(n,x/2,1/2) of order 1/2.
+10
12
1, 3, -1, 15, -10, 1, 105, -105, 21, -1, 945, -1260, 378, -36, 1, 10395, -17325, 6930, -990, 55, -1, 135135, -270270, 135135, -25740, 2145, -78, 1, 2027025, -4729725, 2837835, -675675, 75075, -4095, 105, -1, 34459425, -91891800, 64324260, -18378360, 2552550, -185640, 7140, -136
COMMENTS
These polynomials appear in the radial l=0 (s) wave functions of the isotropic three-dimensional harmonic quantum oscillator with the dimensionless variable x=(r/L)^2 with r>=0 and L^2=h/(m*f0). h is Planck's constant and m and f0 are the mass and the frequency of the oscillator.
See A099174 for relations to the Hermite polynomials and the link in A176230 for operator relations. The infinitesimal generator for this matrix contains A014105. The row polynomials are P(n,x) = 2^n n! Lag(n,x/2,1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q, with raising operator R = -x^(-2) [x^(3/2) (1 - 2D)]^2 = 3 - x + (4x - 6) D - 4x D^2 with D = d/dx, i.e., R P(n,x) - P(n+1,x). A matrix reresentation of R acting on an o.g.f. (formal power series) is given by the transpose of the production matrix below. The diagonal corresponds to (3 + 4 xD) x^n = (3 + 4n) x^n; the upper diagonal, to -x x^n = -x^(n+1); and the lower diagonal, to (-6 - 4 xD) D x^n = -n (6 + 4(n-1)) x^(n-1), the sequence A002943. See A176230 for a similar relation. (End)
Exponential Riordan array [1/(1-2x)^(3/2), -x/(1-2x)]. - Paul Barry, Mar 07 2017
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
a(n,m) = n!*(2^(n-m))*L(1/2,n,m) with L(1/2,n,m) = ((-1)^m)*binomial(n+1/2,n-m)/m!, n >= m >= 0, otherwise 0.
Let IP be the lower triangular matrix with its first subdiagonal equal to the first subdiagonal (cf. A014105) of this entry's unsigned matrix M and with all other elements equal to zero. Then IP is the infinitesimal generator of M, i.e., M = exp(IP). - Tom Copeland, Dec 12 2015 Production matrix is
3, -1;
-6, 7, -1;
0, -20, 11, -1;
0, 0, -42, 15, -1;
0, 0, 0, -72, 19, -1;
0, 0, 0, 0, -110, 23, -1;
0, 0, 0, 0, 0, -156, 27, -1;
0, 0, 0, 0, 0, 0, -210, 31, -1;
0, 0, 0, 0, 0, 0, 0, -272, 35, -1;
... (End)
EXAMPLE
[1]; [3,-1]; [15,-10,1]; [105,-105,21,-1]; [945,-1260,378,-36,1]; ...
MAPLE
seq(seq(n!*2^(n-m)*(-1)^m*binomial(n+1/2, n-m)/m!, m=0..n), n=0..10); # Robert Israel, Dec 25 2015
MATHEMATICA
Table[n! (2^(n - m)) ((-1)^m) Binomial[n + 1/2, n - m]/m!, {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Dec 24 2015 *)
CROSSREFS
Cf. A021009 (Coefficient table of n!*L(n, 0, x)). Cf. A001147, A002943, A014105, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299, A176230.
A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).
+10
11
1, 1, 1, 0, 3, 1, 0, 3, 6, 1, 0, 0, 15, 10, 1, 0, 0, 15, 45, 15, 1, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0, 0, 0, 0, 10395, 62370
COMMENTS
a(n,1) = A019590(n) = A008279(1,n). a(n,m) =: S1(-1; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))* A001497(n-1,m-1) (signed Bessel triangle). The monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference). Exponential Riordan array [1+x, x(1+x/2)]. T(n,k) = A001498(k+1, n-k). - Paul Barry, Jan 15 2009
FORMULA
a(n, m) = n!* A030528(n, m)/(m!*2^(n-m)) for n >= m >= 1. a(n, m) = (2*m-n+1)*a(n-1, m) + a(n-1, m-1) for n >= m >= 1 with a(n, m) = 0 for n < m, a(n, 0) := 0, and a(1, 1) = 1. [The 0th column does not appear in this array. - Petros Hadjicostas, Oct 28 2019] E.g.f. for the m-th column: (x*(1 + x/2))^m/m!.
EXAMPLE
Triangle a(n,m) (with rows n >= 1 and columns m >= 1) begins as follows:
1; with row polynomial E(1,x) = x;
1, 1; with row polynomial E(2,x) = x^2 + x;
0, 3, 1; with row polynomial E(3,x) = 3*x^2 + x^3;
0, 3, 6, 1; with row polynomial E(4,x) = 3*x^2 + 6*x^3 + x^4;
0, 0, 15, 10, 1;
0, 0, 15, 45, 15, 1;
0, 0, 0, 105, 105, 21, 1;
0, 0, 0, 105, 420, 210, 28, 1;
...
MAPLE
# The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n<2, 1, 0), 9); # Peter Luschny, Jan 28 2016
MATHEMATICA
t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 11}, {k, n}] // Flatten
(* Second program: *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 13;
M = BellMatrix[If[#<2, 1, 0]&, rows];
Triangle T(n,k) by rows: number of ways k dominoes can be placed on an n X n chessboard, k>=0.
+10
11
1, 1, 1, 4, 2, 1, 12, 44, 56, 18, 1, 24, 224, 1044, 2593, 3388, 2150, 552, 36, 1, 40, 686, 6632, 39979, 157000, 407620, 695848, 762180, 510752, 192672, 35104, 2180, 1, 60, 1622, 26172, 281514, 2135356, 11785382, 48145820, 146702793, 333518324, 562203148
COMMENTS
Also, coefficients of the matching-generating polynomial of the n X n grid graph.
In the n-th row there are floor(n^2/2)+1 values.
EXAMPLE
Triangle starts:
1
1
1 4 2
1 12 44 56 18
1 24 224 1044 2593 3388 2150 552 36
1 40 686 6632 39979 157000 407620 695848 762180 510752 192672 35104 2180
...
MAPLE
b:= proc(n, l) option remember; local k;
if n=0 then 1
elif min(l[])>0 then b(n-1, map(h->h-1, l))
else for k while l[k]>0 do od; expand(`if`(n>1,
x*b(n, subsop(k=2, l)), 0) +`if`(k<nops(l) and l[k+1]=0,
x*b(n, subsop(k=1, k+1=1, l)), 0) +b(n, subsop(k=1, l)))
fi
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$n])):
MATHEMATICA
b[n_, l_List] := b[n, l] = Module[{k}, Which[n == 0, 1, Min[l]>0, b[n-1, l-1], True, For[k=1, l[[k]]>0, k++]; Expand[If[n>1, x*b[n, ReplacePart[l, k -> 2]], 0] + If[k<Length[l] && l[[k+1]] == 0, x*b[n, ReplacePart[l, {k -> 1, k + 1 -> 1}]], 0] + b[n, ReplacePart[l, k -> 1]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, Array[0&, n]]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 16 2015, after Alois P. Heinz *)
PROG
(Sage)
def T(n, k):
G = Graph(graphs.Grid2dGraph(n, n))
G.relabel()
mu = G.matching_polynomial()
return abs(mu[n^2-2*k])
Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].
+10
6
1, 1, 1, 3, 6, 1, 15, 45, 15, 1, 105, 420, 210, 28, 1, 945, 4725, 3150, 630, 45, 1, 10395, 62370, 51975, 13860, 1485, 66, 1, 135135, 945945, 945945, 315315, 45045, 3003, 91, 1, 2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1, 34459425
COMMENTS
Row sums are A066223. Reverse of A119743. Inverse is alternating sign version. Diagonal sums are essentially A025164. See A099174 for relations to the Hermite polynomials and the link for operator relations, including the infinitesimal generator containing A000384. Row polynomials are 2^n n! Lag(n,-x/2,-1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q.
Divided along the diagonals by the initial element ( A001147) of the diagonal, this matrix becomes the even rows of A034839. (End)
The first few rows appear in expansions related to the Dedekind eta function on pp. 537-538 of the Chan et al. link. - Tom Copeland, Dec 14 2016
FORMULA
Number triangle T(n,k) = (2n)!/((2k)!(n-k)!2^(n-k)).
[x^(1/2)(1+2D)]^2 p(n,x)= p(n+1,x) and [D/(1+2D)]p(n,x)= n p(n-1,x) for the row polynomials of T, with D=d/dx. - Tom Copeland, Dec 26 2012 E.g.f.: exp[t*x/(1-2x)]/(1-2x)^(1/2). - Tom Copeland , Dec 10 2013 The n-th row polynomial R(n,x) is given by the type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} (2*k+1)*(2*k+3)*...*(2*k+1+2*(n-1))*(x/2)^k/k!. Cf. A113278. - Peter Bala, Jun 23 2014 The raising operator in my 2012 formula expanded is R = [x^(1/2)(1+2D)]^2 = 1 + x + (2 + 4x) D + 4x D^2, which in matrix form acting on an o.g.f. (formal power series) is the transpose of the production array below. The linear term x is the diagonal of ones after transposition. The main diagonal comes from (1 + 4xD) x^n = (1 + 4n) x^n. The last diagonal comes from (2 D + 4 x D^2) x^n = (2 + 4 xD) D x^n = n * (2 + 4(n-1)) x^(n-1). - Tom Copeland, Dec 13 2015 T(n, k) = (-2)^(n-k)*[x^k] KummerU(-n, 1/2, x). - Peter Luschny, Jan 18 2020
EXAMPLE
Triangle begins
1,
1, 1,
3, 6, 1,
15, 45, 15, 1,
105, 420, 210, 28, 1,
945, 4725, 3150, 630, 45, 1,
10395, 62370, 51975, 13860, 1485, 66, 1,
135135, 945945, 945945, 315315, 45045, 3003, 91, 1,
2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1
Production matrix is
1, 1,
2, 5, 1,
0, 12, 9, 1,
0, 0, 30, 13, 1,
0, 0, 0, 56, 17, 1,
0, 0, 0, 0, 90, 21, 1,
0, 0, 0, 0, 0, 132, 25, 1,
0, 0, 0, 0, 0, 0, 182, 29, 1,
0, 0, 0, 0, 0, 0, 0, 240, 33, 1.
MAPLE
ser := n -> series(KummerU(-n, 1/2, x), x, n+1):
seq(seq((-2)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8); # Peter Luschny, Jan 18 2020
MATHEMATICA
t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); u[n_, k_] := t[2 n, k + n]; Table[ u[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jan 14 2011 *)
CROSSREFS
Cf. A000384, A001147, A034839, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299.
Triangular array read by rows: T(n, k) = S(n, [n/2]-k) and S(n,k) = C(n, 2*k)*(2*k-1)!!*((2*k-1)!! + 1)/2, n>=0, 0<=k<=[n/2].
+10
1
1, 1, 1, 1, 3, 1, 6, 6, 1, 30, 10, 1, 120, 90, 15, 1, 840, 210, 21, 1, 5565, 3360, 420, 28, 1, 50085, 10080, 756, 36, 1, 446985, 250425, 25200, 1260, 45, 1, 4916835, 918225, 55440, 1980, 55, 1, 54033210, 29501010, 2754675, 110880, 2970, 66, 1
FORMULA
T(n, k) = (n!/(j!*(n-2*j)!))*(2^(-j-1)+Gamma(j+1/2)/sqrt(4*Pi)) where j = floor(n/2) - k.
EXAMPLE
Triangle starts:
[ 0] 1,
[ 1] 1,
[ 2] 1, 1,
[ 3] 3, 1,
[ 4] 6, 6, 1,
[ 5] 30, 10, 1,
[ 6] 120, 90, 15, 1,
[ 7] 840, 210, 21, 1,
[ 8] 5565, 3360, 420, 28, 1,
[ 9] 50085, 10080, 756, 36, 1,
[10] 446985, 250425, 25200, 1260, 45, 1.
MAPLE
T := proc(n, k) local j; j := iquo(n, 2) - k;
(n!/(j!*(n-2*j)!))*(2^(-j-1)+GAMMA(j+1/2)/sqrt(4*Pi)) end:
seq(print(seq(T(n, k), k=0..iquo(n, 2))), n=0..10);
MATHEMATICA
row[n_] := FunctionExpand[HypergeometricPFQ[{-n/2, (1-n)/2}, {}, 2z] + HypergeometricPFQ[{1/2, -n/2, (1-n)/2}, {}, 4z]]/2 // CoefficientList[#, z]& // Reverse;
PROG
(Sage)
from sage.functions.hypergeometric import closed_form
R.<z> = ZZ[]
h = hypergeometric([-n/2, (1-n)/2], [], 2*z)
g = hypergeometric([1/2, -n/2, (1-n)/2], [], 4*z)
T = R(((closed_form(h)+closed_form(g))/2)).coefficients()
return T[::-1]
a(n) is the number of zeros of the Hermite H(n, x) polynomial in the open interval (-1, +1).
+10
0
0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9
COMMENTS
The Hermite polynomials satisfy the following recurrence relation:
H(0,x) = 1,
H(1,x) = 2*x,
H(n,x) = 2*x*H(n-1,x) - 2*(n-1)*H(n-2,x).
The first few Hermite polynomials are:
H(0,x) = 1
H(1,x) = 2x
H(2,x) = 4x^2 - 2
H(3,x) = 8x^3 - 12x
H(4,x) = 16x^4 - 48x^2 + 12
H(5,x) = 32x^5 - 160x^3 + 120x
EXAMPLE
a(5) = 3 because the zeros of H(5,x) = 32x^5 - 160x^3 + 120x are x1 = -2.0201828..., x2 = -.9585724..., x3 = 0., x4 = .9585724... and x5 = 2.020182... with three roots x2, x3 and x4 in the open interval (-1, +1).
MAPLE
for n from 0 to 90 do:it:=0:
y:=[fsolve(expand(HermiteH(n, x)), x, real)]:for m from 1 to nops(y) do:if abs(y[m])<1 then it:=it+1:else fi:od: printf(`%d, `, it):od:
MATHEMATICA
a[n_] := Length@ List@ ToRules@ Reduce[ HermiteH[n, x] == 0 && -1 < x < 1, x]; Array[a, 82, 0] (* Giovanni Resta, May 17 2017 *)
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