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%I #27 Jan 04 2021 08:44:04
%S 1,2,37,1000,38401,1896876,112124629,7679202336,595411451905,
%T 51348552829300,4861414171762501,500163335120177136,
%U 55466421261812540929,6585829687114412247800,832587068884779776276661,111541424966889778569909376,15771414153994526723881828353
%N a(n) = Sum_{k=0..n} n^k * binomial(n,k)^4.
%C In general, Sum_{k=0..n} n^k * binomial(n,k)^p is asymptotic to (1+n^(1/p))^(n*p+p-1) / sqrt(p * (2*Pi)^(p-1) * n^(p-1/p)).
%H Vincenzo Librandi, <a href="/A234971/b234971.txt">Table of n, a(n) for n = 0..200</a> [a(0)=1 inserted by _Georg Fischer_, Jan 04 2020]
%H Vaclav Kotesovec, <a href="http://www.kotesovec.cz/math_articles/kotesovec_binomial_asymptotics.pdf">Asymptotic of a sums of powers of binomial coefficients * x^k</a>, 2012
%F a(n) ~ (1+n^(1/4))^(4*n+3) / (4*sqrt(2) * Pi^(3/2) * n^(15/8)).
%F a(n) = hypergeom([-n, -n, -n, -n], [1, 1, 1], n). - _Peter Luschny_, Dec 22 2020
%p a := n -> hypergeom([-n, -n, -n, -n], [1, 1, 1], n):
%p seq(simplify(a(n)), n=0..16); # _Peter Luschny_, Dec 22 2020
%t Table[Sum[If[n==k==0, 1, n^k]*Binomial[n, k]^4, {k, 0, n}], {n, 0, 20}] (* offset adapted by _Georg Fischer_, Jan 04 2021 *)
%o (PARI) a(n) = sum(k=0, n, n^k * binomial(n,k)^4); \\ _Michel Marcus_, Jan 04 2021
%Y Cf. A187021, A241247.
%K nonn,easy
%O 0,2
%A _Vaclav Kotesovec_, Apr 19 2014
%E a(0) = 1 prepended by _Peter Luschny_, Dec 22 2020
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