A234971 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^4.

1, 2, 37, 1000, 38401, 1896876, 112124629, 7679202336, 595411451905, 51348552829300, 4861414171762501, 500163335120177136, 55466421261812540929, 6585829687114412247800, 832587068884779776276661, 111541424966889778569909376, 15771414153994526723881828353

Offset: 0

Author

Vaclav Kotesovec, Apr 19 2014

Keywords

• nonn
• easy

Comments

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)).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200 [a(0)=1 inserted by _Georg Fischer_, Jan 04 2020]
• Vaclav Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012

Crossrefs

Cf. A187021, A241247.

Programs

Maple

a := n -> hypergeom([-n, -n, -n, -n], [1, 1, 1], n):
seq(simplify(a(n)), n=0..16); # Peter Luschny, Dec 22 2020

Mathematica

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 *)

PARI

a(n) = sum(k=0, n, n^k * binomial(n,k)^4); \\ Michel Marcus, Jan 04 2021

Formula

a(n) ~ (1+n^(1/4))^(4*n+3) / (4*sqrt(2) * Pi^(3/2) * n^(15/8)).

a(n) = hypergeom([-n, -n, -n, -n], [1, 1, 1], n). - Peter Luschny, Dec 22 2020

Extensions

a(0) = 1 prepended by Peter Luschny, Dec 22 2020