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%I #80 Jul 06 2020 20:44:29
%S 1,-2,1,7,-8,-65,37,1024,1351,-19001,-92618,232513,4087189,9953926,
%T -123909155,-1170342533,-676144160,62840385619,490129709977,
%U -551829062288,-40624407525941,-305175084654341,698633855671510,34571970743398621,278738497423756153,-663168571756087538
%N Expansion of e.g.f. exp(3 * (1 - exp(x)) + x).
%F a(0) = 1 and a(n) = a(n-1) - 3 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
%F a(n) = exp(3) * Sum_{k>=0} (k + 1)^n * (-3)^k / k!.
%F a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -3). - _Vaclav Kotesovec_, Jul 06 2020
%t m = 25; Range[0, m]! * CoefficientList[Series[Exp[3 * (1 - Exp[x]) + x], {x, 0, m}], x] (* _Amiram Eldar_, Jul 06 2020 *)
%t Table[Sum[Binomial[n, k] * BellB[k, -3], {k, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, Jul 06 2020 *)
%o (PARI) N=40; x='x+O('x^N); Vec(serlaplace(exp(3*(1-exp(x))+x)))
%Y Column k=3 of A335977.
%Y Cf. A078940, A335981.
%K sign
%O 0,2
%A _Seiichi Manyama_, Jul 06 2020
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