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%I #17 Jan 15 2016 11:32:05
%S 1,5,109,3533,133501,5629505,254899765,12129399245,599084606845,
%T 30455459491505,1584249399505609,83970120618566825,
%U 4520585403820052581,246592348286170615097,13603606921687170927109,757808346139996787715533,42575668004558257371188605,2410024012619343278147357297
%N Sum_{k = 0..n} binomial(n,k)^3*binomial(n+k,k)^2.
%C Compare with the Apéry numbers A005258 and A005259.
%H Vincenzo Librandi, <a href="/A014180/b014180.txt">Table of n, a(n) for n = 0..200</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of generalized Apery sequences with powers of binomial coefficients</a>, Nov 04 2012
%F 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
%F 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
%t Table[Sum[Binomial[n,k]^3*Binomial[n+k,k]^2,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Nov 04 2012 *)
%o (PARI) a(n)=sum(k=0,n, binomial(n,k)^3*binomial(n+k,k)^2 ); \\ _Joerg Arndt_, May 04 2013
%Y Cf. A218693, A112019, A111968, A014178, A218689, A218692, A005258, A005259.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_.
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