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%I #10 Jul 15 2024 08:33:54
%S 0,1,2,5,16,56,218,937,4376,22027,118744,681570,4144988,26598313,
%T 179451366,1268930969,9378332608,72267300476,579336907254,
%U 4822070246225,41597773001612,371306237988959,3424303740576440,32583334570211654,319487530199710232,3224337031346853361
%N Expansion of e.g.f. sin(x) * exp(exp(x) - 1).
%C The first negative term is a(71).
%F a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * Bell(n-2*k-1).
%F Conjecture: a(n) = (i/(2*e))*Sum_{k=0..oo} ((k - i)^n - (k + i)^n)/(k!), where i = sqrt(-1) and e = exp(1). - _Velin Yanev_, Jul 06 2024
%t nmax = 25; CoefficientList[Series[Sin[x] Exp[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[(-1)^k Binomial[n, 2 k + 1] BellB[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}], {n, 0, 25}]
%Y Cf. A000110, A009542, A102286, A351745.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Apr 07 2022
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