A059304 a(n) = 2^n * (2*n)! / (n!)^2.
1, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720, 24893440, 189190144, 1444724736, 11076222976, 85201715200, 657270374400, 5082890895360, 39392404439040, 305870434467840, 2378992268083200, 18531097667174400
Offset: 0
Links
- Harry J. Smith, Table of n, a(n) for n = 0..200
- Paul Barry and Arnauld Mesinga Mwafise, Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.
- Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
- Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
- H. J. Brothers, Pascal's Prism: Supplementary Material.
- Tucker J. Ervin, Blake Jackson, Jay Lane, Kyungyong Lee, Son Dang Nguyen, Jack O'Donohue and Michael Vaughan, Permutations whose Reverse Shares the Same Recording Tableau in the Robinson-Schensted Correspondence, Séminaire Lotharingien de Combinatoire 86 (2022), Article B86a.
- Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
- Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Crossrefs
Programs
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Magma
[2^n*Binomial(2*n,n): n in [0..25]]; // Vincenzo Librandi, Oct 08 2015
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Maple
seq(binomial(2*n,n)*2^n,n=0..19); # Zerinvary Lajos, Dec 08 2007
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Mathematica
Table[2^n Binomial[2n,n],{n,0,30}] (* Harvey P. Dale, Dec 16 2014 *) -
PARI
{a(n)=if(n<0, 0, 2^n*(2*n)!/n!^2)} /* Michael Somos, Jan 31 2007 */ -
PARI
{ for (n = 0, 200, write("b059304.txt", n, " ", 2^n * (2*n)! / n!^2); ) } \\ Harry J. Smith, Jun 25 2009 -
PARI
/* as lattice paths: same as in A092566 but use */ steps=[[1, 0], [1, 0], [0, 1]]; /* note the double [1, 0] */ /* Joerg Arndt, Jul 01 2011 */
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SageMath
def A059304(n): return pow(2,n)*binomial(2*n,n) print([A059304(n) for n in range(41)]) # G. C. Greubel, Jan 18 2025
Formula
a(n) = 2^n * C(2*n,n).
D-finite with recurrence a(n) = 4*(2-1/n)*a(n-1).
G.f.: 1/sqrt(1-8*x) - T. D. Noe, Jun 11 2002
E.g.f.: exp(4*x)*BesselI(0, 4*x). - Vladeta Jovovic, Aug 20 2003
a(n) = A038207(n,n). - Joerg Arndt, Jul 01 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 4*x*(2*k+1)/(4*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
E.g.f.: E(0)/2, where E(k) = 1 + 1/(1 - 4*x/(4*x + (k+1)^2/(2*k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: Q(0)/(1+2*sqrt(x)), where Q(k) = 1 + 2*sqrt(x)/(1 - 2*sqrt(x)*(2*k+1)/(2*sqrt(x)*(2*k+1) + (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 09 2013
O.g.f.: hypergeom([1/2], [], 8*x). - Peter Luschny, Oct 08 2015
a(n) = Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n,k)*binomial(3*n-2*k,n)* binomial(n+k,n). - Peter Bala, Aug 04 2016
a(n) ~ 8^n/sqrt(Pi*n). - Ilya Gutkovskiy, Aug 04 2016
From Amiram Eldar, Jul 21 2020: (Start)
Sum_{n>=0} 1/a(n) = 8/7 + 8*sqrt(7)*arcsin(1/sqrt(8))/49.
Sum_{n>=0} (-1)^n/a(n) = (8/27)*(3 - arcsinh(1/sqrt(8))). (End)
a(n) = Sum_{k = n..2*n} binomial(2*n,k)*binomial(k,n). In general, for m >= 1, Sum_{k = n..m*n} binomial(m*n,k)*binomial(k,n) = 2^((m-1)*n)*binomial(m*n,n). - Peter Bala, Mar 25 2023
Conjecture: a(n) = Sum_{0 <= j, k <= n} binomial(n, j)*binomial(n, k)* binomial(k+j, n). - Peter Bala, Jul 16 2024
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