A346181 a(n) = Sum_{k=0..n} binomial(n,k) * k^n * (k+1)^(n-1), with a(0)=1.

1, 1, 16, 660, 55360, 7898080, 1720310784, 532246677760, 222180859015168, 120419919713396736, 82254072931642408960, 69147812281254588276736, 70171320441471903522619392, 84590682101275469336895029248, 119503325912427987607346212765696, 195568947175627297113759888436101120

Offset

0,3

Links

Table of n, a(n) for n=0..15.

Formula

a(n) ~ 2^(2*n - 1/2) * n^(2*n - 1) / (sqrt(1+c) * c^(n + 1/2) * (2+c)^(n-1) * exp(2*n)), where c = LambertW(2*exp(-2)) = 0.21771510575709011079475830443312...

Maple

A346181 := proc(n)
    add(binomial(n, k)*k^n*(k+1)^(n-1), k=0..n) ;
end proc:
seq(A346181(n), n=0..30) ; # R. J. Mathar, Mar 02 2023

Mathematica

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

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k) * k^n * (k+1)^(n-1)); \\ Michel Marcus, Jul 09 2021

Crossrefs

Cf. A217900.

Sequence in context: A334734 A220298 A222916 * A197407 A197429 A231936

Adjacent sequences: A346178 A346179 A346180 * A346182 A346183 A346184

Keyword

nonn

Author

Vaclav Kotesovec, Jul 09 2021

Status

approved