A308864 a(n) = Sum_{k>=0} (n*k + 1)^n/2^(k+1).

1, 2, 17, 442, 22833, 1942026, 245246761, 43001877122, 9986424563009, 2965574161158490, 1095862246322273601, 493067173454342315346, 265360795458419332828657, 168311426029488910748596394, 124248479512164840358578103577, 105608722927065949313865618984226

Offset: 0

Author

Ilya Gutkovskiy, Jun 29 2019

Keywords

• nonn

Crossrefs

Cf. A000629, A000670, A080253, A285067, A307066.

Programs

Mathematica

Table[Sum[(n k + 1)^n/2^(k + 1), {k, 0, Infinity}], {n, 0, 15}]
Table[n! SeriesCoefficient[Exp[x]/(2 - Exp[n x]), {x, 0, n}], {n, 0, 15}]
Join[{1}, Table[Sum[Binomial[n, k] n^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 1, 15}]]

Formula

a(n) = n! * [x^n] exp(x)/(2 - exp(n*x)).

a(n) = Sum_{k=0..n} binomial(n,k) * n^k * A000670(k).

a(n) ~ sqrt(Pi/2) * n^(2*n + 1/2) / (log(2)^(n+1) * exp(n)). - Vaclav Kotesovec, Jun 29 2019