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%I #75 Jul 06 2020 20:44:50
%S 1,-3,5,5,-43,-27,597,805,-11883,-40475,265685,2133157,-3405803,
%T -107760283,-301542315,4458255397,42421260949,-45046794011,
%U -3365690666283,-19844416105563,138274174035221,2917746747446373,11092963732101461,-207438902364296411,-3205301465165742187
%N Expansion of e.g.f. exp(4 * (1 - exp(x)) + x).
%F a(0) = 1 and a(n) = a(n-1) - 4 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
%F a(n) = exp(4) * Sum_{k>=0} (k + 1)^n * (-4)^k / k!.
%F a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -4). - _Vaclav Kotesovec_, Jul 06 2020
%t m = 24; Range[0, m]! * CoefficientList[Series[Exp[4 * (1 - Exp[x]) + x], {x, 0, m}], x] (* _Amiram Eldar_, Jul 06 2020 *)
%t Table[Sum[Binomial[n, k] * BellB[k, -4], {k, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, Jul 06 2020 *)
%o (PARI) N=40; x='x+O('x^N); Vec(serlaplace(exp(4*(1-exp(x))+x)))
%Y Column k=4 of A335977.
%Y Cf. A078945, A335982.
%K sign
%O 0,2
%A _Seiichi Manyama_, Jul 06 2020
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