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A234971 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^4. 4
1, 2, 37, 1000, 38401, 1896876, 112124629, 7679202336, 595411451905, 51348552829300, 4861414171762501, 500163335120177136, 55466421261812540929, 6585829687114412247800, 832587068884779776276661, 111541424966889778569909376, 15771414153994526723881828353 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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]
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
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 *)
PROG
(PARI) a(n) = sum(k=0, n, n^k * binomial(n, k)^4); \\ Michel Marcus, Jan 04 2021
CROSSREFS
Sequence in context: A307318 A058245 A257995 * A139108 A165697 A320994
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Apr 19 2014
EXTENSIONS
a(0) = 1 prepended by Peter Luschny, Dec 22 2020
STATUS
approved

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Last modified March 1 02:07 EST 2024. Contains 370429 sequences. (Running on oeis4.)