Displaying 1-10 of 41 results found.
1, 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904
Central Delannoy numbers: a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k).
(Formerly M2942 N1184)
+10
190
1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823, 252055236609, 1425834724419, 8079317057869, 45849429914943, 260543813797441, 1482376214227923, 8443414161166173, 48141245001931263
COMMENTS
Number of paths from (0,0) to (n,n) in an n X n grid using only steps north, northeast and east (i.e., steps (1,0), (1,1), and (0,1)).
Also the number of ways of aligning two sequences (e.g., of nucleotides or amino acids) of length n, with at most 2*n gaps (-) inserted, so that while unnecessary gappings: - -a a- - are forbidden, both b- and -b are allowed. (If only other of the latter is allowed, then the sequence A000984 gives the number of alignments.) There is an easy bijection from grid walks given by Dickau to such set of alignments (e.g., the straight diagonal corresponds to the perfect alignment with no gaps). - Antti Karttunen, Oct 10 2001 Also main diagonal of array A008288 defined by m(i,1) = m(1,j) = 1, m(i,j) = m(i-1,j-1) + m(i-1,j) + m(i,j-1). - Benoit Cloitre, May 03 2002 So, as a special case of Dmitry Zaitsev's Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= n from any given point. These terms occur in the crystal ball sequences: a(n) here is the n-th term in the sequence for the n-dimensional cubic lattice. See A008288 for a list of crystal ball sequences (rows or columns of A008288). - Shel Kaphan, Dec 26 2022 a(n) is the number of n-matchings of a comb-like graph with 2*n teeth. Example: a(2) = 13 because the graph consisting of a horizontal path ABCD and the teeth Aa, Bb, Cc, Dd has 13 2-matchings: any of the six possible pairs of teeth and {Aa, BC}, {Aa, CD}, {Bb, CD}, {Cc, AB}, {Dd, AB}, {Dd, BC}, {AB, CD}. - Emeric Deutsch, Jul 02 2002 Number of ordered trees with 2*n+1 edges, having root of odd degree, nonroot nodes of outdegree at most 2 and branches of odd length. - Emeric Deutsch, Aug 02 2002 The sum of the first n coefficients of ((1 - x) / (1 - 2*x))^n is a(n-1). - Michael Somos, Sep 28 2003 Also number of paths from (0,0) to (n,0) using only steps U = (1,1), H = (1,0) and D =(1,-1), U can have 2 colors and H can have 3 colors. - N-E. Fahssi, Jan 27 2008 Number of overpartitions in the n X n box (treat a walk of the type in the first comment as an overpartition, by interpreting a NE step as N, E with the part thus created being overlined). - William J. Keith, May 19 2017 Diagonal of rational functions 1/(1 - x - y - x*y), 1/(1 - x - y*z - x*y*z). - Gheorghe Coserea, Jul 03 2018 Dimensions of endomorphism algebras End(R^{(n)}) in the Delannoy category attached to the oligomorphic group of order preserving self-bijections of the real line. - Noah Snyder, Mar 22 2023
REFERENCES
Frits Beukers, Arithmetic properties of Picard-Fuchs equations, Séminaire de Théorie des nombres de Paris, 1982-83, Birkhäuser Boston, Inc.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 593.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Math., 26 (1961), 223-229.
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, Wadsworth, Vol. 2, 1999; see Example 6.3.8 and Problem 6.49.
D. B. West, Combinatorial Mathematics, Cambridge, 2021, p. 28.
LINKS
Cyril Banderier and Sylviane Schwer, Why Delannoy numbers?, Journal of Statistical Planning and Inference, 135(1) (2005), 40-54.
FORMULA
a(n) = P_n(3), where P_n is n-th Legendre polynomial.
G.f.: 1 / sqrt(1 - 6*x + x^2).
Dominant term in asymptotic expansion is binomial(2*n, n)/2^(1/4)*((sqrt(2) + 1)/2)^(2*n + 1)*(1 + c_1/n + c_2/n^2 + ...). - Michael David Hirschhorna(n) = Sum_{i=0..n} ( A000079(i)* A008459(n, i)) = Sum_{i=0..n} (2^i * C(n, i)^2). - Antti Karttunen, Oct 10 2001 a(n) = Sum_{k=0..n} C(n+k, n-k)*C(2*k, k). - Benoit Cloitre, Feb 13 2003 a(n) = Sum_{k=0..n} C(n, k)^2 * 2^k. - Michael Somos, Oct 08 2003 G.f. of a(n-1) = 1 / (1 - x / (1 - 2*x / (1 - 2*x / (1 - x / (1 - 2*x / (1 - x / ...)))))). - Michael Somos, May 11 2012 a(n) = Sum_{k=0..n} C(2*n-k, n)*C(n, k). - Paul Barry, Apr 23 2005 D-finite with recurrence: a(-1) = a(0) = 1; n*a(n) = 3*(2*n-1)*a(n-1) - (n-1)*a(n-2). Eq (4) in T. D. Noe's article in JIS 9 (2006) #06.2.7. Define general Delannoy numbers by (i,j > 0): d(i,0) = d(0,j) = 1 =: d(0,0) and d(i,j) = d(i-1,j-1) + d(i-2,j-1) + d(i-1,j). Then a(k) = Sum_{j >= 0} d(k,j)^2 + d(k-1,j)^2 = A026933(n)+ A026933(n-1). This is a special case of the following formula for general Delannoy numbers: d(k,j) = Sum_{i >= 0, p=0..n} d(p, i) * d(n-p, j-i) + d(p-1, i) * d(n-p-1, j-i-1). - Peter E John, Oct 19 2006 Coefficient of x^n in (1 + 3*x + 2*x^2)^n. - N-E. Fahssi, Jan 11 2008 G.f.: 1/(1 - x - 2*x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - ... (continued fraction). - Paul Barry, May 28 2009 G.f.: 1/(2*Q(0) + x - 1) where Q(k) = 1 + k*(1-x) - x - x*(k + 1)*(k + 2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013 a(n) = Sum_{k=0..n} C(n,k) * C(n+k,k). - Joerg Arndt, May 11 2013 G.f.: G(0), where G(k) = 1 + x*(6 - x)*(4*k + 1)/(4*k + 2 - 2*x*(6-x)*(2*k + 1)*(4*k + 3)/(x*(6 - x)*(4*k + 3) + 4*(k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2013 G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(6 - x)*(2*k - 1)/(x*(6 - x)*(2*k - 1) + 2*(k + 1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(6 - x)*(2*k + 1)/(x*(6 - x)*(2*k + 1) + 2*(k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013 a(n) = Sum_{k=0..n/2} C(n-k,k) * 3^(n-2*k) * 2^k * C(n,k). - Vladimir Kruchinin, Jun 29 2015 a(n) = Sum_{0 <= j, k <= n} (-1)^(n+j)*C(n,k)*C(n,j)*C(n+k,k)*C(n+k+j,k+j). Cf. A126086 and A274668. - Peter Bala, Jan 15 2020 a(n) ~ c * (3 + 2*sqrt(2))^n / sqrt(n), where c = 1/sqrt(4*Pi*(3*sqrt(2)-4)) = 0.572681... (Banderier and Schwer, 2005). - Amiram Eldar, Jun 07 2020 a(n+1) = 3*a(n) + 2*Sum_{l=1..n} A006318(l)*a(n-l). [Eq. (1.16) in Qi-Shi-Guo (2016)] a(n) ~ (1 + sqrt(2))^(2*n+1) / (2^(5/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 09 2023
EXAMPLE
G.f. = 1 + 3*x + 13*x^2 + 63*x^3 + 321*x^4 + 1683*x^5 + 8989*x^6 + ...
MAPLE
seq(add(multinomial(n+k, n-k, k, k), k=0..n), n=0..20); # Zerinvary Lajos, Oct 18 2006
MATHEMATICA
f[n_] := Sum[ Binomial[n, k] Binomial[n + k, k], {k, 0, n}]; Array[f, 21, 0] (* Or *)
a[0] = 1; a[1] = 3; a[n_] := a[n] = (3(2 n - 1)a[n - 1] - (n - 1)a[n - 2])/n; Array[a, 21, 0] (* Or *)
a[n_] := Hypergeometric2F1[-n, n+1, 1, -1]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 26 2013 *) a[ n_] := With[ {m = If[n < 0, -1 - n, n]}, SeriesCoefficient[ (1 - 6 x + x^2)^(-1/2), {x, 0, m}]]; (* Michael Somos, Jun 10 2015 *)
PROG
(PARI) {a(n) = if( n<0, n = -1 - n); polcoeff( 1 / sqrt(1 - 6*x + x^2 + x * O(x^n)), n)}; /* Michael Somos, Sep 23 2006 */ (PARI) {a(n) = if( n<0, n = -1 - n); subst( pollegendre(n), x, 3)}; /* Michael Somos, Sep 23 2006 */ (PARI) {a(n) = if( n<0, n = -1 - n); n++; subst( Pol(((1 - x) / (1 - 2*x) + O(x^n))^n), x, 1); } /* Michael Somos, Sep 23 2006 */ (PARI) a(n)=if(n<0, 0, polcoeff((1+3*x+2*x^2)^n, n)) \\ Paul Barry, Aug 22 2007 (PARI) /* same as in A092566 but use */ steps=[[1, 0], [0, 1], [1, 1]]; /* Joerg Arndt, Jun 30 2011 */ (PARI) a(n)=sum(k=0, n, binomial(n, k)*binomial(n+k, k)); \\ Joerg Arndt, May 11 2013 (PARI) x='x+O('x^100); Vec(1/sqrt(1 - 6*x + x^2)) \\ Altug Alkan, Oct 17 2015 (Python) # from Nick Hobson.
def f(a, b):
if a == 0 or b == 0:
return 1
return f(a, b - 1) + f(a - 1, b) + f(a - 1, b - 1)
[f(n, n) for n in range(7)]
(Python)
from gmpy2 import divexact
for n in range(2, 10**3):
(Maxima) a(n):=coeff(expand((1+3*x+2*x^2)^n), x, n);
(Sage)
a = lambda n: hypergeometric([-n, -n], [1], 2)
CROSSREFS
Cf. A008288, bisection of A026003, A027618, A047665, A052141, A084773, A152250, A109980, A000129, A078057, A241023, A243949.
EXTENSIONS
New name and reference Sep 15 1995
Formula and more references from Don Knuth, May 15 1996
Decimal expansion of Pi/4.
+10
93
7, 8, 5, 3, 9, 8, 1, 6, 3, 3, 9, 7, 4, 4, 8, 3, 0, 9, 6, 1, 5, 6, 6, 0, 8, 4, 5, 8, 1, 9, 8, 7, 5, 7, 2, 1, 0, 4, 9, 2, 9, 2, 3, 4, 9, 8, 4, 3, 7, 7, 6, 4, 5, 5, 2, 4, 3, 7, 3, 6, 1, 4, 8, 0, 7, 6, 9, 5, 4, 1, 0, 1, 5, 7, 1, 5, 5, 2, 2, 4, 9, 6, 5, 7, 0, 0, 8, 7, 0, 6, 3, 3, 5, 5, 2, 9, 2, 6, 6, 9, 9, 5, 5, 3, 7
COMMENTS
Also the ratio of the area of a circle to the circumscribed square. More generally, the ratio of the area of an ellipse to the circumscribed rectangle. Also the ratio of the volume of a cylinder to the circumscribed cube. - Omar E. Pol, Sep 25 2013 Also the surface area of a quarter-sphere of diameter 1. - Omar E. Pol, Oct 03 2013 Dirichlet L-series of the non-principal character modulo 4 ( A101455) at 1. See e.g. Table 22 of arXiv:1008.2547. - R. J. Mathar, May 27 2016 This constant is also equal to the infinite sum of the arctangent functions with nested radicals consisting of square roots of two. Specifically, one of the Viete-like formulas for Pi is given by Pi/4 = Sum_{k = 2..oo} arctan(sqrt(2 - a_{k - 1})/a_k), where the nested radicals are defined by recurrence relations a_k = sqrt(2 + a_{k - 1}) and a_1 = sqrt(2) (see the article [Abrarov and Quine]). - Sanjar Abrarov, Jan 09 2017 Pi/4 is the area enclosed between circumcircle and incircle of a regular polygon of unit side. - Mohammed Yaseen, Nov 29 2023
REFERENCES
Jörg Arndt and Christoph Haenel, Pi: Algorithmen, Computer, Arithmetik, Springer 2000, p. 150.
Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books, p. 408.
J. Rivaud, Analyse, Séries, équations différentielles, Mathématiques supérieures et spéciales, Premier cycle universitaire, Vuibert, 1981, Exercice 3, p. 136.
FORMULA
Equals Integral_{x=0..oo} sin(2x)/(2x) dx.
Equals Integral_{x=0..Pi/2} sin(x)^2 dx, or Integral_{x=0..Pi/2} cos(x)^2 dx. - Jean-François Alcover, Mar 26 2013 Equals (Sum_{x=0..oo} sin(x)*cos(x)/x) - 1/2. - Bruno Berselli, May 13 2013 Equals Product_{k in A071904} (if k mod 4 = 1 then (k-1)/(k+1)) else (if k mod 4 = 3 then (k+1)/(k-1)). - Dimitris Valianatos, Oct 05 2016 For N even: 2*(Pi/4 - Sum_{k = 1..N/2} (-1)^(k-1)/(2*k - 1)) ~ (-1)^(N/2)*(1/N - 1/N^3 + 5/N^5 - 61/N^7 + 1385/N^9 - ...), where the sequence of unsigned coefficients [1, 1, 5, 61, 1385, ...] is A000364. See Borwein et al., Theorem 1 (a). For N odd: 2*(Pi/4 - Sum_{k = 1..(N-1)/2} (-1)^(k-1)/(2*k - 1)) ~ (-1)^((N-1)/2)*(1/N - 1/N^2 + 2/N^4 - 16/N^6 + 272/N^8 - ...), where the sequence of unsigned coefficients [1, 1, 2, 16, 272, ...] is A000182 with an extra initial term of 1. For N = 0,1,2,... and m = 1,3,5,... there holds Pi/4 = (2*N)! * m^(2*N) * Sum_{k >= 0} ( (-1)^(N+k) * 1/Product_{j = -N..N} (2*k + 1 + 2*m*j) ); when N = 0 we get the Madhava-Gregory-Leibniz series for Pi/4.
For examples of asymptotic expansions for the tails of these series representations for Pi/4 see A024235 (case N = 1, m = 1), A278080 (case N = 2, m = 1) and A278195 (case N = 3, m = 1). For N = 0,1,2,..., Pi/4 = 4^(N-1)*N!/(2*N)! * Sum_{k >= 0} 2^(k+1)*(k + N)!* (k + 2*N)!/(2*k + 2*N + 1)!, follows by applying Euler's series transformation to the above series representation for Pi/4 in the case m = 1. (End)
For k = 0,1,2,..., Pi/4 = k!*Sum_{n = -oo..oo} 1/((4*n+1)*(4*n+3)* ...*(4*n+2*k+1)), where Sum_{n = -oo..oo} f(n) is understood as lim_{j -> oo} Sum_{n = -j..j} f(n).
Equals Integral_{x = 0..oo} sin(x)^4/x^2 dx = Sum_{n >= 1} sin(n)^4/n^2, by the Abel-Plana formula.
Equals Integral_{x = 0..oo} sin(x)^3/x dx = Sum_{n >= 1} sin(n)^3/n, by the Abel-Plana formula. (End)
Equals arcsin(1/sqrt(2)).
Equals Product_{k>=1} (1 - 1/(2*k+1)^2).
Equals Integral_{x=0..oo} x/(x^4 + 1) dx.
Equals Integral_{x=0..oo} 1/(x^2 + 4) dx. (End)
Equals Integral_{x = 0..oo} exp(-x)*sin(x)/x dx (see Rivaud reference). - Bernard Schott, Jan 28 2022 Equals beta(1), where beta is the Dirichlet beta function.
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p)^(-1). (End)
Equals arctan( F(1)/F(4) ) + arctan( F(2)/F(3) ), where F(1), F(2), F(3), and F(4) are any four consecutive Fibonacci numbers. - Gary W. Adamson, Mar 03 2024 Pi/4 = Sum_{n >= 1} i/(n*P(n, i)*P(n-1, i)) = (1/2)*Sum_{n >= 1} (-1)^(n+1)*4^n/(n* A006139(n)* A006139(n-1)), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. The n-th summand of the series is O( 1/(3 + 2*sqrt(3))^n ). - Peter Bala, Mar 16 2024 Equals arctan( phi^(-3) ) + arctan(phi^(-1) ). - Gary W. Adamson, Mar 27 2024
EXAMPLE
0.785398163397448309615660845819875721049292349843776455243736148...
N = 2, m = 6: Pi/4 = 4!*3^4 Sum_{k >= 0} (-1)^k/((2*k - 11)*(2*k - 5)*(2*k + 1)*(2*k + 7)*(2*k + 13)). - Peter Bala, Nov 15 2016
MATHEMATICA
(* PROGRAM STARTS *)
(* Define the nested radicals a_k by recurrence *)
a[k_] := Nest[Sqrt[2 + #1] & , 0, k]
(* Example of Pi/4 approximation at K = 100 *)
Print["The actual value of Pi/4 is"]
N[Pi/4, 40]
Print["At K = 100 the approximated value of Pi/4 is"]
K := 100; (* the truncating integer *)
N[Sum[ArcTan[Sqrt[2 - a[k - 1]]/a[k]], {k, 2, K}], 40] (* equation (8) *)
(* Error terms for Pi/4 approximations *)
Print["Error terms for Pi/4"]
k := 1; (* initial value of the index k *)
K := 10; (* initial value of the truncating integer K *)
sqn := {}; (* initiate the sequence *)
AppendTo[sqn, {"Truncating integer K ", " Error term in Pi/4"}];
While[K <= 30,
AppendTo[sqn, {K,
N[Pi/4 - Sum[ArcTan[Sqrt[2 - a[k - 1]]/a[k]], {k, 2, K}], 1000] //
N}]; K++]
Print[MatrixForm[sqn]]
PROG
(Haskell) -- see link: Literate Programs
import Data.Char (digitToInt)
a003881_list len = map digitToInt $ show $ machin `div` (10 ^ 10) where
machin = 4 * arccot 5 unity - arccot 239 unity
unity = 10 ^ (len + 10)
arccot x unity = arccot' x unity 0 (unity `div` x) 1 1 where
arccot' x unity summa xpow n sign
| term == 0 = summa
| otherwise = arccot'
x unity (summa + sign * term) (xpow `div` x ^ 2) (n + 2) (- sign)
where term = xpow `div` n
(SageMath) # Leibniz/Cohen/Villegas/Zagier/Arndt/Haenel
def FastLeibniz(n):
b = 2^(2*n-1); c = b; s = 0
for k in range(n-1, -1, -1):
t = 2*k+1
s = s + c/t if is_even(k) else s - c/t
b *= (t*(k+1))/(2*(n-k)*(n+k))
c += b
return s/c
A003881 = RealField(3333)(FastLeibniz(1330))
(Magma) R:= RealField(100); Pi(R)/4; // G. C. Greubel, Mar 08 2018
a(n) = 2^((n-1)*(n+2)/2).
+10
29
1, 4, 32, 512, 16384, 1048576, 134217728, 34359738368, 17592186044416, 18014398509481984, 36893488147419103232, 151115727451828646838272, 1237940039285380274899124224, 20282409603651670423947251286016, 664613997892457936451903530140172288
COMMENTS
Number of redundant paths for a fault-tolerant ATM switch.
Hankel transform (see A001906 for definition ) of A001850, A006139, A084601; also Hankel transform of the sequence 1, 0, 4, 0, 24, 0, 160, 0, 1120, ... ( A059304 with interpolated zeros). - Philippe Deléham, Jul 03 2005 a(n) = the multiplicative Wiener index of the wheel graph with n+3 vertices. The multiplicative Wiener index of a connected simple graph G is defined as the product of the distances between all pairs of distinct vertices of G. The wheel graph with n+3 vertices has (n+3)(n+2)/2 pairs of distinct vertices, of which 2(n+2) are adjacent; each of the remaining (n+2)(n-1)/2 pairs are at distance 2; consequently, the multiplicative Wiener index is 2^((n-1)(n+2)/2) = a(n). - Emeric Deutsch, Aug 17 2015
PROG
(Magma) I:=[1]; [n le 1 select I[n] else Self(n-1)*2^n: n in [1..20]]; // Vincenzo Librandi, Oct 24 2012 (Maxima) A036442[n]:=2^((n-1)*(n+2)/2)$
AUTHOR
Abdallah Rayhan (rayhan(AT)engr.uvic.ca)
a(n+2) = (2n+3)*a(n+1) + (n+1)^2*a(n), a(0) = 1, a(1) = 1.
+10
13
1, 1, 4, 24, 204, 2220, 29520, 463680, 8401680, 172504080, 3958113600, 100370793600, 2787459998400, 84139894238400, 2742857884166400, 96034297911552000, 3594206259195552000, 143193586818810528000, 6050501147565883008000, 270263264589232282368000
COMMENTS
a(n) is the number of n-letter words from an n-letter alphabet such that no letter appears more than twice. - Paul Boddington, Nov 17 2003
FORMULA
a(n) = n!/2^n* A006139(n) = n!*Sum_{k=floor(n/2)..n} 2^(k-n)*C(n, k)*C(k, n-k). Sum_{n>=0} a(n)*x^n/n!^2 = exp(x)*BesselI(0, sqrt(2)*x). a(n) is the central coefficient of n!*(1+x+x^2/2)^n. - Vladeta Jovovic, Mar 22 2004 The function B(x) := int {t=0..x} A(t), obtained by integrating the generating function A(x), satisfies the autonomous differential equation d/dx(B(x)) = 1/(cos(B(x))-sin(B(x))). Compare with A190392. Thus B(x), and hence A(x), can be found by inverting the function int {t=0..x} (cos(t)-sin(t)). By applying [Dominici, Theorem 4.1] the result can be expressed as
A(x) = 1 + sum {n>=1} D^n[1/(cos(t)-sin(t))](0)*x^n/n!, where the nested derivative D^n[f](x) of a function f(x) is defined recursively as D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0. Thus a(n) = D^n[1/(cos(t)-sin(t))](0). (End)
E.g.f. at offset 1: Series_Reversion(cos(x) + sin(x) - 1). - Paul D. Hanna, Aug 08 2012 a(n) ~ (1+sqrt(2))^(n+1/2) * n^n / (2^(1/4) * exp(n)). - Vaclav Kotesovec, Feb 18 2017
MAPLE
f := proc(n) option remember; if n <= 1 then 1 else (2*n-1)*f(n-1) +(n-1)^2*f(n-2); fi; end;
MATHEMATICA
Range[0, 20]! CoefficientList[Series[1/(1-2x-x^2)^(1/2), {x, 0, 20}], x] (* Geoffrey Critzer, Dec 07 2011 *)
PROG
(PARI) {a(n)=local(X=x+x^2*O(x^n)); (n+1)!*polcoeff(serreverse(cos(X)+sin(X)-1), n+1)} \\ Paul D. Hanna, Aug 08 2012
Expansion of 1/sqrt(1-4*x^2-4*x^3).
+10
8
1, 0, 2, 2, 6, 12, 26, 60, 130, 300, 672, 1540, 3514, 8064, 18552, 42756, 98802, 228624, 530024, 1230372, 2860000, 6655792, 15505932, 36159552, 84398626, 197154984, 460903796, 1078251044, 2524144224, 5912535672, 13857378300, 32495267712
COMMENTS
Diagonal sums of number triangle A115951. Number of lattice paths from (0,0) to (n,n) using steps (2,1), (1,0), (1,2). - Joerg Arndt, Jul 05 2011 Diagonal of rational function 1/(1 - (x^2 + y^2 + x^3*y)). - Seiichi Manyama, Mar 22 2023
FORMULA
a(n) = Sum_{k=0..floor(n/2)} C(2*k,k)*C(k,n-2*k).
G.f.: Q(0), where Q(k) = 1 + 4*x*(x+x^2)*(4*k+1) / (4*k+2 - 4*x*(x+x^2)*(4*k+2)*(4*k+3) / (4*x*(x+x^2)*(4*k+3) + 4*(k+1) / Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013 D-finite with recurrence: n*a(n) - 4*(n-1)*a(n-2) - 2*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jan 14 2020
MAPLE
option remember;
if n < 4 then
op(n+1, [1, 0, 2, 2]);
else
4*(n-1)*procname(n-2)+2*(2*n-3)*procname(n-3) ;
%/n ;
end if;
end proc:
MATHEMATICA
CoefficientList[Series[1/Sqrt[1-4x^2-4x^3], {x, 0, 35}], x] (* or *) Table[Sum[Binomial[2k, k] Binomial[k, n-2k], {k, 0, Floor[n/2]}], {n, 0, 35}] (* Michael De Vlieger, Sep 03 2015 *)
PROG
(PARI) x = xx+O(xx^40); Vec(1/sqrt(1-4*x^2-4*x^3)) \\ Michel Marcus, Sep 03 2015 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1-4*x^2-4*x^3) )); // G. C. Greubel, May 06 2019 (Sage) (1/sqrt(1-4*x^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 06 2019
Expansion of 1/sqrt(1 - 4*x/(1+x)^3).
+10
7
1, 2, 0, -4, -4, 6, 18, 4, -48, -70, 60, 288, 170, -686, -1386, 432, 4928, 4806, -9684, -27572, -3672, 84106, 118162, -122388, -537834, -284830, 1386840, 2688944, -1103362, -10181934, -9354198, 21404728, 57921144, 3663942, -185437360, -248708676, 292137656
FORMULA
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+2*k-1,n-k).
n*a(n) = -( -2*a(n-1) + (2*n)*a(n-2) + 4*(n-3)*a(n-3) + (n-4)*a(n-4) ) for n > 3.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+1-k,2) * a(k).
a(n) = (-1)^(n+1)*n*(n + 1)*hypergeom([3/2, 1-n, 1+n/2, (3+n)/2], [4/3, 5/3, 2], 2^4/3^3) for n > 0. - Stefano Spezia, Jul 11 2024
MATHEMATICA
a[n_]:=(-1)^(n+1)n(n+1)HypergeometricPFQ[{3/2, 1-n, 1+n/2, (3+n)/2}, {4/3, 5/3, 2}, 2^4/3^3]; Join[{1}, Array[a, 36]] (* Stefano Spezia, Jul 11 2024 *)
PROG
(PARI) my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^3))
a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * binomial(2*(n-3*k),n-3*k).
+10
7
1, 2, 6, 20, 72, 264, 984, 3712, 14136, 54224, 209200, 810912, 3155616, 12320512, 48239232, 189336192, 744722400, 2934759360, 11584470336, 45796087680, 181285742592, 718498695424, 2850802065152, 11322567705600, 45011437903104, 179088911779328
COMMENTS
Diagonal of rational function 1/(1 - (x + y + x^4*y^3)). - Seiichi Manyama, Mar 23 2023
FORMULA
G.f.: 1/sqrt(1 - 4*x*(1 + x^3)).
n*a(n) = 2*(2*n-1)*a(n-1) + 2*(2*n-4)*a(n-4).
a(n) ~ 1 / (2*sqrt((1 - 3*r)*Pi*n) * r^n), where r = 0.2463187933841190115229... is the positive real root of the equation -1 + 4*r + 4*r^4 = 0. - Vaclav Kotesovec, Mar 23 2023
MATHEMATICA
Table[Sum[Binomial[n-3k, k]Binomial[2(n-3k), n-3k], {k, 0, Floor[n/3]}], {n, 0, 30}] (* Harvey P. Dale, May 27 2024 *)
PROG
(PARI) a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k));
(PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x^3)))
Expansion of 1/sqrt(1 - 4*x/(1+x)^4).
+10
7
1, 2, -2, -8, 6, 42, -8, -228, -90, 1210, 1238, -6116, -10864, 28574, 80932, -116248, -548010, 339678, 3455686, 173208, -20452674, -14036418, 113365140, 156407916, -580805472, -1312098918, 2659610562, 9621079540, -9902139124, -64566648122, 18521111032
FORMULA
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+3*k-1,n-k).
n*a(n) = -( (n-3)*a(n-1) + (6*n-6)*a(n-2) + 10*(n-3)*a(n-3) + 5*(n-4)*a(n-4) + (n-5)*a(n-5) ) for n > 4.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+2-k,3) * a(k).
a(n) = (-1)^(n+1)*Pochhammer(n,3)*hypergeom([1-n, 1+n/3, (4+n)/3, (5+n)/3], [5/4, 7/4, 2], 3^3/2^6)/3 for n > 0. - Stefano Spezia, Jul 11 2024
MATHEMATICA
a[n_]:=(-1)^(n+1)Pochhammer[n, 3]HypergeometricPFQ[{1-n, 1+n/3, (4+n)/3, (5+n)/3}, {5/4, 7/4, 2}, 3^3/2^6]/3; Join[{1}, Array[a, 30]] (* Stefano Spezia, Jul 11 2024 *)
PROG
(PARI) my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^4))
(PARI) a(n)=sum(k=0, n, (-1)^(n-k) * binomial(2*k, k) * binomial(n+3*k-1, n-k)) \\ Winston de Greef, Mar 24 2023
Expansion of 1/sqrt(1 - 4*x/(1+x)^5).
+10
7
1, 2, -4, -10, 30, 72, -238, -580, 1970, 4910, -16734, -42750, 144600, 379000, -1264700, -3402480, 11160730, 30828070, -99168820, -281279030, 885931600, 2580541580, -7948885910, -23779051760, 71572652480, 219906488302, -646332447086, -2039738985238, 5850898295170
FORMULA
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+4*k-1,n-k).
n*a(n) = -( (2*n-4)*a(n-1) + (11*n-14)*a(n-2) + 20*(n-3)*a(n-3) + 15*(n-4)*a(n-4) + 6*(n-5)*a(n-5) + (n-6)*a(n-6) ) for n > 5.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+3-k,4) * a(k).
a(n) = (-1)^(n+1)*Pochhammer(n,4)*hypergeom([3/2, 1-n, 1+n/4, (5+n)/4, (6+n)/4, (7+n)/4], [6/5, 7/5, 8/5, 9/5, 2], 2^10/5^5)/12 for n > 0. - Stefano Spezia, Jul 11 2024
MATHEMATICA
a[n_]:=(-1)^(n+1)Pochhammer[n, 4]HypergeometricPFQ[{3/2, 1-n, 1+n/4, (5+n)/4, (6+n)/4, (7+n)/4}, {6/5, 7/5, 8/5, 9/5, 2}, 2^10/5^5]/12; Join[{1}, Array[a, 28]] (* Stefano Spezia, Jul 11 2024 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^5))
(PARI) a(n) = sum(k=0, n, (-1)^(n-k) * binomial(2*k, k) * binomial(n+4*k-1, n-k)) \\ Winston de Greef, Mar 24 2023
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