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Sum_{k=0..n} C(n,k)^4*C(n+k,k)^2.
4

%I #12 Sep 12 2015 11:00:28

%S 1,5,181,9797,616501,45778505,3678753205,314352091205,28154344881205,

%T 2612340741982505,249415797821914681,24374175871569953225,

%U 2428502737999745059381,245952659136011117909897,25260608441078125508484181,2626023605592369669830217797

%N Sum_{k=0..n} C(n,k)^4*C(n+k,k)^2.

%H Vincenzo Librandi, <a href="/A218690/b218690.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+sqrt(5))/2)^(2*(5*n+4)-3/2)/(5^(1/4)*(2*Pi*n)^(5/2)*sqrt(2)).

%t Table[Sum[Binomial[n,k]^4*Binomial[n+k,k]^2,{k,0,n}],{n,0,20}]

%Y Cf. A005258, A218692.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Nov 04 2012