A014180 Sum_{k = 0..n} binomial(n,k)^3*binomial(n+k,k)^2.

1, 5, 109, 3533, 133501, 5629505, 254899765, 12129399245, 599084606845, 30455459491505, 1584249399505609, 83970120618566825, 4520585403820052581, 246592348286170615097, 13603606921687170927109, 757808346139996787715533, 42575668004558257371188605, 2410024012619343278147357297

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Compare with the Apéry numbers A005258 and A005259.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• V. Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

Crossrefs

Cf. A218693, A112019, A111968, A014178, A218689, A218692, A005258, A005259.

Programs

Mathematica

Table[Sum[Binomial[n,k]^3*Binomial[n+k,k]^2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Nov 04 2012 *)

PARI

a(n)=sum(k=0,n, binomial(n,k)^3*binomial(n+k,k)^2 ); \\ Joerg Arndt, May 04 2013

Formula

a(n) ~ (1+r)^(4*n+5/2)/r^(5*n+9/2)/(4*Pi^2*n^2)*sqrt((1-r)/(5+r)), where r is positive real root of the equation (1-r)^3*(1+r)^2 = r^5, r = 0.65039847669867... - Vaclav Kotesovec, Nov 04 2012

The expansions exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 5*x + 67*x^2 + 1471*x^3 + 41456*x^4 + 1380268*x^5 + ... and exp( Sum_{n >= 1} a(n-1)*x^n/n ) = 1 + x + 3*x^2 + 39*x^3 + 924*x^4 + 27696*x^5 + ... appear to have integer coefficients. Cf. A005258 and A005259.- Peter Bala, Jan 14 2016