A036909 a(n) = (2/3) * 4^n * binomial(3*n, n).

8, 160, 3584, 84480, 2050048, 50692096, 1270087680, 32133218304, 819082035200, 21002987765760, 541167892561920, 13999778090188800, 363391162981023744, 9459706464902840320, 246865719056498950144, 6456334894356662059008, 169176689745174567321600, 4440485304168581976555520

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• Identity (3.116) in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 35.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..695

Crossrefs

Cf. A005809, A025174.

Programs

Magma

[(2/3)*4^n*Binomial(3*n,n): n in [1..30]]; // G. C. Greubel, Jun 22 2022

Mathematica

Table[2/3 4^n Binomial[3n,n],{n,20}](* Harvey P. Dale, Mar 26 2012 *)

SageMath

[(2/3)*4^n*binomial(3*n, n) for n in (0..30)] # G. C. Greubel, Jun 22 2022

Formula

Sum_{k=0..n} binomial(4*n, 2*(n-k))*binomial(k+n, n) = (2/3)*4^n*binomial(3*n, n) = (2/3)*4^n*A005809(n) = 2*4^n*A025174(n).

G.f.: (2/3) * 2F1([1/3, 2/3], [1/2], 27*x) = 2*(cos((1/6)*arccos(1-54*x))/sqrt(1-27*x) - 1) /(3*x). - Harvey P. Dale, Mar 26 2012

D-finite with recurrence n*(2*n-1)*a(n) = 6*(3*n-1)*(3*n-2)*a(n-1). - R. J. Mathar, Feb 08 2021

G.f.: (2/3)*(cos((1/3)*Arcsin(3*sqrt(3*x)))/sqrt(1-27*x) - 1). - G. C. Greubel, Jun 22 2022

a(n) ~ 3^(3*n)/sqrt(3*n*Pi). - Stefano Spezia, Apr 25 2024