A320258 - OEIS

%I #5 Oct 08 2018 18:15:51

%S 1,1,-3,10,81,-4724,156205,-4406814,76958273,3775676248,-698309272899,

%T 72802616429830,-6310377003297455,435451735391849892,

%U -10028808876450831571,-4757293711381352201774,1464955115044140633346305,-310063138309576689774123728,55179706013436631385620675837

%N a(n) = n! * [x^n] exp(x*exp(-n*x)).

%F a(n) = [x^n] Sum_{k>=0} x^k/(1 + n*k*x)^(k+1).

%F a(n) = Sum_{k=0..n} binomial(n,k)*(-n*k)^(n-k).

%t Table[n! SeriesCoefficient[Exp[x Exp[-n x]], {x, 0, n}], {n, 0, 18}]

%t Table[SeriesCoefficient[Sum[x^k/(1 + n k x)^(k + 1), {k, 0, n}], {x, 0, n}], {n, 0, 18}]

%t Join[{1}, Table[Sum[Binomial[n, k] (-n k)^(n - k), {k, 0, n}], {n, 18}]]

%Y Cf. A003725, A295552.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Oct 08 2018