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%I #7 Jun 29 2019 08:58:33
%S 1,2,17,442,22833,1942026,245246761,43001877122,9986424563009,
%T 2965574161158490,1095862246322273601,493067173454342315346,
%U 265360795458419332828657,168311426029488910748596394,124248479512164840358578103577,105608722927065949313865618984226
%N a(n) = Sum_{k>=0} (n*k + 1)^n/2^(k+1).
%F a(n) = n! * [x^n] exp(x)/(2 - exp(n*x)).
%F a(n) = Sum_{k=0..n} binomial(n,k) * n^k * A000670(k).
%F a(n) ~ sqrt(Pi/2) * n^(2*n + 1/2) / (log(2)^(n+1) * exp(n)). - _Vaclav Kotesovec_, Jun 29 2019
%t Table[Sum[(n k + 1)^n/2^(k + 1), {k, 0, Infinity}], {n, 0, 15}]
%t Table[n! SeriesCoefficient[Exp[x]/(2 - Exp[n x]), {x, 0, n}], {n, 0, 15}]
%t Join[{1}, Table[Sum[Binomial[n, k] n^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 1, 15}]]
%Y Cf. A000629, A000670, A080253, A285067, A307066.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jun 29 2019
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