| Displaying 1-6 of 6 results found.
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1
Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A126931.
+20
0
1, 3, 1, 10, 6, 1, 33, 29, 9, 1, 110, 126, 57, 12, 1, 366, 518, 306, 94, 15, 1, 1220, 2052, 1494, 600, 140, 18, 1, 4065, 7925, 6849, 3389, 1035, 195, 21, 1, 13550, 30030, 30025, 17628, 6635, 1638, 259, 24, 1
FORMULA
T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + Sum_{i>=0} T(n-1,k+1+i)*(-3)^i. - Philippe Deléham, Feb 23 2012
EXAMPLE
Triangle begins:
1 ;
3,1 ;
10,6,1 ;
33,29,9,1 ;
110,126,57,12,1 ; ...
Fredholm-Rueppel sequence.
+10
121
1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
COMMENTS
Binary representation of the Kempner-Mahler number Sum_{k>=0} 1/2^(2^k) = A007404. Also a(n) = A(n) mod 2 where A is any of A001700, A005573, A007854, A026641, A049027, A064063, A064088, A064090, A064092, A064325, A064327, A064329, A064331, A064613, A076026, A105523, A123273, A126694, A126930, A126931, A126982, A126983, A126987, A127016, A127053, A127358, A127360, A127361, A127363. - Philippe Deléham, May 26 2007 a(n) = (product of digits of n; n in binary notation) mod 2. This sequence is a transformation of the Thue-Morse sequence ( A010060), since there exists a function f such that f(sum of digits of n) = (product of digits of n). - Ctibor O. Zizka, Feb 12 2008 a(n-1), n >= 1, the characteristic sequence for powers of 2, A000079, is the unique solution of the following formal product and formal power series identity: Product_{j>=1} (1 + a(j-1)*x^j) = 1 + Sum_{k>=1} x^k = 1/(1-x). The product is therefore Product_{l>=1} (1 + x^(2^l)). Proof. Compare coefficients of x^n and use the binary representation of n. Uniqueness follows from the recurrence relation given for the general case under A147542. - Wolfdieter Lang, Mar 05 2009 a(n) is also the number of orbits of length n for the map x -> 1-cx^2 on [-1,1] at the Feigenbaum critical value c=1.401155... . - Thomas Ward, Apr 08 2009 For n >= 2, also the largest exponent k >= 0 such that n^k in binary notation does not contain both 0 and 1. Unlike for the decimal version of this sequence, A062518, where the terms are only conjectural, for this sequence the values of a(n) can be proved to be the characteristic function of A000225, as follows: n^k will contain both 0 and 1 unless n^k = 2^r-1 for some r. But this is a special case of Catalan's equation x^p = y^q-1, which was proved by Preda Mihăilescu to have no nontrivial solution except 2^3 = 3^2 - 1. - Christopher J. Smyth, Aug 22 2014 Image, under the coding a,b -> 1; c -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cb, c -> cc. - Jeffrey Shallit, May 14 2016 Number of nonisomorphic Boolean algebras of order n+1. - Jianing Song, Jan 23 2020
LINKS
D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999. Eric Rowland and Reem Yassawi, Profinite automata, arXiv:1403.7659 [math.DS], 2014. See p. 8.
FORMULA
1 followed by a string of 2^k - 1 0's. Also a(n)=1 iff n = 2^m - 1.
Sum_{n>=0} 1/10^(2^n) = 0.110100010000000100000000000000010...
1 if n=0, floor(log_2(n+1)) - floor(log_2(n)) otherwise. G.f.: (1/x) * Sum_{k>=0} x^(2^k) = Sum_{k>=0} x^(2^k-1). - Ralf Stephan, Apr 28 2003 Dirichlet g.f. for right-shifted sequence: 2^(-s)/(1-2^(-s)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*Sum_{j=0..k} binomial(k, 2^j-1). - Paul Barry, Jun 01 2006 a(n) = Sum_{k=1..n} A191898(n,k)*cos(Pi*(n-1)*(k-1))/n; (conjecture). - Mats Granvik, Mar 04 2013 a(n) = 1 iff n=2^k-1 for some k, 0 otherwise. - M. F. Hasler, Jun 20 2014 a(n) = ceiling(log_2(n+2)) - ceiling(log_2(n+1)). - Gionata Neri, Sep 06 2015 a(n) = ( A000913(n-1) mod 2), for n>1. a(n) = ( A000917(n-1) mod 2), for n>0. a(n) = ( A002057(n-2) mod 2), for n>1. a(n) = floor(1+log(n+1)/log(2)) - floor(log(2n+1)/log(2)). - Adriano Caroli, Sep 22 2019 This is also the decimal expansion of -Sum_{k>=1} mu(2*k)/(10^k - 1), where mu is the Möbius function ( A008683). - Amiram Eldar, Jul 12 2020
EXAMPLE
G.f. = 1 + x + x^3 + x^7 + x^15 + x^31 + x^63 + x^127 + x^255 + x^511 + ...
a(7) = 1 since 7 = 2^3 - 1, while a(10) = 0 since 10 is not of the form 2^k - 1 for any integer k.
MAPLE
A036987:= n-> `if`(2^ilog2(n+1) = n+1, 1, 0):
MATHEMATICA
RealDigits[ N[ Sum[1/10^(2^n), {n, 0, Infinity}], 110]][[1]]
(* Recurrence: *)
t[n_, 1] = 1; t[1, k_] = 1;
t[n_, k_] := t[n, k] =
If[n < k, If[n > 1 && k > 1, -Sum[t[k - i, n], {i, 1, n - 1}], 0],
If[n > 1 && k > 1, Sum[t[n - i, k], {i, 1, k - 1}], 0]];
Table[t[n, k], {k, n, n}, {n, 104}]
mb2d[n_]:=1 - Module[{n2 = IntegerDigits[n, 2]}, Max[n2] - Min[n2]]; Array[mb2d, 120, 0] (* Vincenzo Librandi, Jul 19 2019 *) Table[PadRight[{1}, 2^k, 0], {k, 0, 7}]//Flatten (* Harvey P. Dale, Apr 23 2022 *)
PROG
(PARI) {a(n) =( n++) == 2^valuation(n, 2)}; /* Michael Somos, Aug 25 2003 */ (Haskell)
a036987 n = ibp (n+1) where
ibp 1 = 1
ibp n = if r > 0 then 0 else ibp n' where (n', r) = divMod n 2
a036987_list = 1 : f [0, 1] where f (x:y:xs) = y : f (x:xs ++ [x, x+y])
-- Same list generator function as for a091090_list, cf. A091090. (Python)
from sympy import catalan
(Python)
CROSSREFS
The first row of A073346. Occurs for first time in A073202 as row 6 (and again as row 8). If interpreted with offset=1 instead of 0 (i.e., a(1)=1, a(2)=1, a(3)=0, a(4)=1, ...) then this is the characteristic function of 2^n ( A000079) and as such occurs as the first row of A073265. Also, in that case the INVERT transform will produce A023359. This is Guy Steele's sequence GS(1, 3), also GS(3, 1) (see A135416).
Catalan triangle (with 0's) read by rows.
+10
109
1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 1, 5, 0, 9, 0, 5, 0, 1, 0, 14, 0, 14, 0, 6, 0, 1, 14, 0, 28, 0, 20, 0, 7, 0, 1, 0, 42, 0, 48, 0, 27, 0, 8, 0, 1, 42, 0, 90, 0, 75, 0, 35, 0, 9, 0, 1, 0, 132, 0, 165, 0, 110, 0, 44, 0, 10, 0, 1, 132, 0, 297, 0, 275, 0, 154, 0, 54, 0, 11, 0
COMMENTS
Inverse lower triangular matrix of A049310(n,m) (coefficients of Chebyshev's S polynomials). Walks with a wall: triangle of number of n-step walks from (0,0) to (n,m) where each step goes from (a,b) to (a+1,b+1) or (a+1,b-1) and the path stays in the nonnegative quadrant.
T(n,m) is the number of left factors of Dyck paths of length n ending at height m. Example: T(4,2)=3 because we have UDUU, UUDU, and UUUD, where U=(1,1) and D=(1,-1). (This is basically a different formulation of the previous - walks with a wall - property.) - Emeric Deutsch, Jun 16 2011 "The Catalan triangle is formed in the same manner as Pascal's triangle, except that no number may appear on the left of the vertical bar." [Conway and Smith]
G.f. for row polynomials p(n,x) := Sum_{m=0..n} (a(n,m)*x^m): c(z^2)/(1-x*z*c(z^2)). Row sums (x=1): A001405 (central binomial). In the language of the Shapiro et al. reference such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. The g.f. Ginv(x) of the m=0 column of the inverse of a given Bell-matrix (here A049310) is obtained from its g.f. of the m=0 column (here G(x)=1/(1+x^2)) by Ginv(x)=(f^{(-1)}(x))/x, with f(x) := x*G(x) and f^{(-1)}is the compositional inverse function of f (here one finds, with Ginv(0)=1, c(x^2)). See the Shapiro et al. reference. Number of involutions of {1,2,...,n} that avoid the patterns 132 and have exactly k fixed points. Example: T(4,2)=3 because we have 2134, 4231 and 3214. Number of involutions of {1,2,...,n} that avoid the patterns 321 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1324 and 2134. Number of involutions of {1,2,...,n} that avoid the patterns 213 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1432 and 4231. - Emeric Deutsch, Oct 12 2006 This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0) -> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007 By columns without the zeros, n-th row = A000108 convolved with itself n times; equivalent to A = (1 + x + 2x^2 + 5x^3 + 14x^4 + ...), then n-th row = coefficients of A^(n+1). - Gary W. Adamson, May 13 2009 As an upper right triangle, rows represent powers of 5-sqrt(24):
5 - sqrt(24)^1 = 0.101020514...
5 - sqrt(24)^2 = 0.010205144...
5 - sqrt(24)^3 = 0.001030928...
(Divided by sqrt(96) these powers give a decimal representation of the columns of A007318, with 1/sqrt(96) being the middle column.) (End) T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) having k (1,0)-steps. Example: T(5,3)=4 because, denoting U=(1,1), D=(1,-1), H=(1,0), we have HHHUD, HHUDH, HUDHH, and UDHHH. - Emeric Deutsch, Jun 01 2011 Let S(N,x) denote the N-th Chebyshev S-polynomial in x (see A049310, cf. [W. Lang]). Then x^n = sum_{k=0..n} T(n,k)*S(k,x). - L. Edson Jeffery, Sep 06 2012 This triangle a(n,m) appears also in the (unreduced) formula for the powers rho(N)^n for the algebraic number over the rationals rho(N) = 2*cos(Pi/N) = R(N, 2), the smallest diagonal/side ratio R in the regular N-gon:
rho(N)^n = sum(a(n,m)*R(N,m+1),m=0..n), n>=0, identical in N >= 1. R(N,j) = S(j-1, x=rho(N)) (Chebyshev S ( A049310)). See a comment on this under A039599 (even powers) and A039598 (odd powers). Proof: see the Sep 06 2012 comment by L. Edson Jeffery, which follows from T(n,k) (called here a(n,k)) being the inverse of the Riordan triangle A049310. - Wolfdieter Lang, Sep 21 2013 The so-called A-sequence for this Riordan triangle of the Bell type (c(x^2), x*c(x^2)) (see comments above) is A(x) = 1 + x^2. This proves the recurrence given in the formula section by Henry Bottomley for a(n, m) = a(n-1, m-1) + a(n-1, m+1) for n>=1 and m>=1, with inputs. The Z-sequence for this Riordan triangle is Z(x) = x which proves the recurrence a(n,0) = a(n-1,1), n>=1, a(0,0) = 1. For A- and Z-sequences for Riordan triangles see the W. Lang link under A006232. - Wolfdieter Lang, Sep 22 2013 Rows of the triangle describe decompositions of tensor powers of the standard (2-dimensional) representation of the Lie algebra sl(2) into irreducibles. Thus a(n,m) is the multiplicity of the m-th ((m+1)-dimensional) irreducible representation in the n-th tensor power of the standard one. - Mamuka Jibladze, May 26 2015 The Riordan row polynomials p(n, x) belong to the Boas-Buck class (see a comment and references in A046521), hence they satisfy the Boas-Buck identity: (E_x - n*1)*p(n, x) = (E_x + 1)*Sum_{j=0..n-1} (1/2)*(1 - (-1)^j)*binomial(j+1, (j+1)/2)*p(n-1-j, x), for n >= 0, where E_x = x*d/dx (Euler operator). For the triangle a(n, m) this entails a recurrence for the sequence of column m, given in the formula section. - Wolfdieter Lang, Aug 11 2017 For row n, the nonzero values represent the odd components (loops) formed by n+1 nonintersecting arches above and below the x-axis with the following constraints: The top has floor((n+3)/2) starting arches at position 1 and the next consecutive odd positions. All other starting top arches are in even positions. The bottom arches are a rainbow of arches. If the component=1 then the arch configuration is a semimeander solution.
Examples: For row 3 {0, 2, 0, 1} there are 3 arch configurations: 2 arch configurations have a component=1; 1 has a component=3. c=components, U=top arch starting in odd position, u=top arch starting in an even position, d=ending top arch:
.
top UuUdUddd c=3 top UdUuUddd c=1 top UdUdUudd c=1
/\ /\
//\\ / \
// \\ / /\ \ /\
// \\ / / \ \ / \
///\ /\\\ /\ / / /\ \ \ /\ /\ / /\ \
\\\ \/ /// \ \ \ \/ / / / \ \ \ \/ / / /
\\\ /// \ \ \ / / / \ \ \ / / /
\\\/// \ \ \/ / / \ \ \/ / /
\\// \ \ / / \ \ / /
\/ \ \/ / \ \/ /
\ / \ /
\/ \/
For row 4 {2, 0, 3, 0, 1} there are 6 arch configurations: 2 have a component=1; 3 have a component=3: 1 has a component=1. (End)
REFERENCES
J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002)
A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.
LINKS
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328. L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
FORMULA
a(n, m) := 0 if n<m or n-m odd, else a(n, m) = (m+1)*binomial(n+1, (n-m)/2)/(n+1);
a(n, m) = (4*(n-1)*a(n-2, m) + 2*(m+1)*a(n-1, m-1))/(n+m+2), a(n, m)=0 if n<m, a(n, -1) := 0, a(0, 0)=1=a(1, 1), a(1, 0)=0.
G.f. for m-th column: c(x^2)*(x*c(x^2))^m, where c(x) = g.f. for Catalan numbers A000108. G.f.: G(t,z) = c(z^2)/(1 - t*z*c(z^2)), where c(z) = (1 - sqrt(1-4*z))/(2*z) is the g.f. for the Catalan numbers ( A000108). - Emeric Deutsch, Jun 16 2011 a(n, m) = a(n-1, m-1) + a(n-1, m+1) if n > 0 and m >= 0, a(0, 0)=1, a(0, m)=0 if m > 0, a(n, m)=0 if m < 0. - Henry Bottomley, Jan 25 2001 Sum_{k>=0} T(m, k)*T(n, k) = 0 if m+n is odd; Sum_{k>=0} T(m, k)*T(n, k) = A000108((m+n)/2) if m+n is even. - Philippe Deléham, May 26 2005 T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i, C(i,j)*(C(i-j,j+k)-C(i-j,j+k+2))}}; Column k has e.g.f. BesselI(k,2x)-BesselI(k+2,2x). - Paul Barry, Feb 16 2006 Sum_{k=0..n} T(n,k)^x = A000027(n+1), A001405(n), A000108(n), A003161(n), A129123(n) for x = 0,1,2,3,4 respectively. - Philippe Deléham, Nov 22 2009 Sum_{k=0..n} T(n,k)*x^k = A126930(n), A126120(n), A001405(n), A054341(n), A126931(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Nov 28 2009 Recurrence for row polynomials C(n, x) := Sum_{m=0..n} a(n, m)*x^m = x*Sum_{k=0..n} Chat(k)*C(n-1-k, x), n >= 0, with C(-1, 1/x) = 1/x and Chat(k) = A000108(k/2) if n is even and 0 otherwise. From the o.g.f. of the row polynomials: G(z; x) := Sum_{n >= 0} C(n, x)*z^n = c(z^2)*(1 + x*z*G(z, x)), with the o.g.f. c of A000108. - Ahmet Zahid KÜÇÜK and Wolfdieter Lang, Aug 23 2015 The Boas-Buck recurrence (see a comment above) for the sequence of column m is: a(n, m) = ((m+1)/(n-m))*Sum_{j=0..n-1-m} (1/2)*(1 - (-1)^j)*binomial(j+1, (j+1)/2)* a(n-1-j, k), for n > m >= 0 and input a(m, m) = 1. - Wolfdieter Lang, Aug 11 2017
EXAMPLE
Triangle a(n,m) begins:
n\m 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 0 1
2: 1 0 1
3: 0 2 0 1
4: 2 0 3 0 1
5: 0 5 0 4 0 1
6: 5 0 9 0 5 0 1
7: 0 14 0 14 0 6 0 1
8: 14 0 28 0 20 0 7 0 1
9: 0 42 0 48 0 27 0 8 0 1
10: 42 0 90 0 75 0 35 0 9 0 1
E.g., the fourth row corresponds to the polynomial p(3,x)= 2*x + x^3.
Production matrix is
0, 1,
1, 0, 1,
0, 1, 0, 1,
0, 0, 1, 0, 1,
0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1, 0, 1 (End)
Boas-Buck recurrence for column k = 2, n = 6: a(6, 2) = (3/4)*(0 + 2*a(4 ,2) + 0 + 6*a(2, 2)) = (3/4)*(2*3 + 6) = 9. - Wolfdieter Lang, Aug 11 2017
MAPLE
T:=proc(n, k): if n+k mod 2 = 0 then (k+1)*binomial(n+1, (n-k)/2)/(n+1) else 0 fi end: for n from 0 to 13 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Oct 12 2006 F:=proc(l, p) if ((l-p) mod 2) = 1 then 0 else (p+1)*l!/( ( (l-p)/2 )! * ( (l+p)/2 +1)! ); fi; end;
r:=n->[seq( F(n, p), p=0..n)]; [seq(r(n), n=0..15)]; # N. J. A. Sloane, Jan 29 2011 A053121 := proc(n, k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1,
procname(n-1, k-1)+procname(n-1, k+1))) end proc:
MATHEMATICA
a[n_, m_] /; n < m || OddQ[n-m] = 0; a[n_, m_] = (m+1) Binomial[n+1, (n-m)/2]/(n+1); Flatten[Table[a[n, m], {n, 0, 12}, {m, 0, n}]] [[1 ;; 90]] (* Jean-François Alcover, May 18 2011 *)
PROG
(Haskell)
a053121 n k = a053121_tabl !! n !! k
a053121_row n = a053121_tabl !! n
a053121_tabl = iterate
(\row -> zipWith (+) ([0] ++ row) (tail row ++ [0, 0])) [1]
(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] + M[n-1, k+1]
return M
(PARI) T(n, m)=if(n<m||(n-m)%2, return(0)); (m+1)*binomial(n+1, (n-m)/2)/(n+1)
CROSSREFS
Variant without zero-diagonals: A033184 and with rows reversed: A009766.
Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k+1) for k >= 1.
+10
24
1, 3, 1, 10, 3, 1, 33, 11, 3, 1, 110, 36, 12, 3, 1, 366, 122, 39, 13, 3, 1, 1220, 405, 135, 42, 14, 3, 1, 4065, 1355, 447, 149, 45, 15, 3, 1, 13550, 4512, 1504, 492, 164, 48, 16, 3, 1, 45162, 15054, 5004, 1668, 540, 180, 51, 17, 3, 1
COMMENTS
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007 Riordan array (2/(1-6x+sqrt(1-4*x^2)),x*c(x^2)) where c(x)= g.f. of the Catalan numbers A000108. - Philippe Deléham, Jun 01 2013
FORMULA
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A126931(m+n).
EXAMPLE
Triangle begins:
1;
3, 1;
10, 3, 1;
33, 11, 3, 1;
110, 36, 12, 3, 1;
366, 122, 39, 13, 3, 1;
1220, 405, 135, 42, 14, 3, 1;
4065, 1355, 447, 149, 45, 15, 3, 1;
MATHEMATICA
T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 3, 0], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)
1, 2, 1, 5, 4, 1, 12, 14, 6, 1, 30, 44, 27, 8, 1, 74, 133, 104, 44, 10, 1, 185, 388, 369, 200, 65, 12, 1, 460, 1110, 1236, 814, 340, 90, 14, 1, 1150, 3120, 3980, 3072, 1560, 532, 119, 16, 1, 2868, 8666, 12432, 10984, 6542, 2715, 784, 152, 18, 1, 7170, 23816, 37938, 37688, 25695, 12516, 4403, 1104, 189, 20, 1
FORMULA
T(n,k) = (2*k*sum(j=0..n+k, binomial(j,-n-3*k+2*j)*binomial(n+k,j)))/(n+k). - Vladimir Kruchinin, Oct 12 2011 T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + Sum_{i>=0} T(n-1,k+1+i)*(-2)^i. - Philippe Deléham, Feb 23 2012
EXAMPLE
The triangle starts
1;
2, 1;
5, 4, 1;
12, 14, 6, 1;
30, 44, 27, 8, 1;
MAPLE
A053121 := proc(n, k) if n< k or type(n-k, 'odd') then 0; else (k+1)*binomial(n+1, (n-k)/2)/(n+1) ; end if; end proc:
A038207 := proc(i, j) if j> i then 0; else binomial(i, j)*2^(i-j) ; end if; end proc:
MATHEMATICA
a53121[n_, k_] /; n < k || OddQ[n - k] = 0;
a53121[n_, k_] := (k + 1) Binomial[n + 1, (n - k)/2]/(n + 1);
a38207[n_, k_] := Sum[Binomial[n, i] Binomial[i, k], {i, 0, n}];
T[n_, k_] := Sum[a53121[n, j] a38207[j, k], {j, 0, n}];
PROG
(Maxima)
T(n, k):=(2*k*sum(binomial(j, -n-3*k+2*j)*binomial(n+k, j), j, 0, n+k))/(n+k); /* Vladimir Kruchinin, Oct 12 2011 */
EXTENSIONS
Edited, definition rewritten, program and more terms added by R. J. Mathar, Mar 25 2010
Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A126930.
+10
0
1, -1, 1, 2, -2, 1, -3, 5, -3, 1, 6, -10, 9, -4, 1, -10, 22, -22, 14, -5, 1, 20, -44, 54, -40, 20, -6, 1, -35, 93, -123, 109, -65, 27, -7, 1, 70, -186, 281, -276, 195, -98, 35, -8, 1, -126, 386, -618, 682, -541, 321, -140, 44, -9, 1
FORMULA
Sum_{k, 0<=k<=n}T(n,k)*x^k = (-1)^n* A126931(n), (-1)^n* A054341(n), A126930(n), A126120(n), A001405(n), A054341(n), A126931(n) for x = -2, -1, 0, 1, 2, 3, 4 respectively.
EXAMPLE
Triangle begins
1
-1, 1
2, -2, 1
-3, 5, -3, 1
6, -10, 9, -4, 1
-10, 22, -22, 14, -5, 1
20, -44, 54, -40, 20, -6, 1
-35, 93, -123, 109, -65, 27, -7, 1
...
Production matrix begins
x, 1
1, x, 1
1, 1, x, 1
1, 1, 1, x, 1
1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, x, 1
..., with x = -1.
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