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A346181
a(n) = Sum_{k=0..n} binomial(n,k) * k^n * (k+1)^(n-1), with a(0)=1.
1
1, 1, 16, 660, 55360, 7898080, 1720310784, 532246677760, 222180859015168, 120419919713396736, 82254072931642408960, 69147812281254588276736, 70171320441471903522619392, 84590682101275469336895029248, 119503325912427987607346212765696, 195568947175627297113759888436101120
OFFSET
0,3
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
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Jul 09 2021
STATUS
approved