A242449 a(n) = Sum_{k=0..n} C(n,k) * (2*k+1)^(2*n+1).

1, 28, 3612, 1064480, 560632400, 462479403072, 550095467201728, 891290348282967040, 1887146395301619304704, 5058811707344107766328320, 16746136671945501439084657664, 67088193422344140016282100785152, 319900900946743851959321101768511488

Offset: 0

Author

Vaclav Kotesovec, May 14 2014

Keywords

• nonn
• easy

Comments

Generally, for p>=1, a(n) = Sum_{k=0..n} C(n,k) * (p*k+1)^(p*n+1) is asymptotic to n^(p*n+1) * p^(p*n+1) * r^(p*n+3/2+1/p) / (sqrt(p+r-p*r) * exp(p*n) * (1-r)^(n+1/p)), where r = p/(p+LambertW(p*exp(-p))).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..190

Crossrefs

Cf. A242373, A220955, A242446, A072034, A007820, A221214, A213193, A195242.

Programs

Mathematica

Table[Sum[Binomial[n,k]*(2*k+1)^(2*n+1),{k,0,n}],{n,0,20}]

PARI

for(n=0,30, print1(sum(k=0,n, binomial(n,k)*(2*k+1)^(2*n+1)), ", ")) \\ G. C. Greubel, Nov 16 2017

Formula

a(n) ~ n^(2*n+1) * 2^(2*n+1) * r^(2*n+2) / (sqrt(2-r) * exp(2*n) * (1-r)^(n+1/2)), where r = 2/(2+LambertW(2*exp(-2))) = 0.901829091937052...