%I #6 Jan 28 2020 04:49:13

%S 1,-1,4,0,-1654,102750,-4079389,-178722208,83191059372,

%T -14561829897345,1115121827539325,403631463559529040,

%U -251989999508801085674,76158421344845152140737,-3994730250899559184766830,-13162858116922635098226480000,10823217968258750568539067678392

%N a(n) = n! * [x^n] exp(n - exp(x) * (exp(n*x) - 1) / (exp(x) - 1)).

%H Vaclav Kotesovec, <a href="/A331582/b331582.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = n! * [x^n] exp(n - exp(x) - exp(2*x) - exp(3*x) - ... - exp(n*x)).

%t Table[n! SeriesCoefficient[Exp[n - Exp[x] (Exp[n x] - 1)/(Exp[x] - 1)], {x, 0, n}], {n, 0, 16}]

%t b[n_, k_] := b[n, k] = If[n == 0, 1, -Sum[Binomial[n - 1, j - 1] Sum[i^j, {i, 1, k}] b[n - j, k], {j, 1, n}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 16}]

%Y Cf. A103438, A319508, A319509, A320288.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Jan 21 2020