A352905 Expansion of e.g.f. sin(x) * exp(exp(x) - 1).
0, 1, 2, 5, 16, 56, 218, 937, 4376, 22027, 118744, 681570, 4144988, 26598313, 179451366, 1268930969, 9378332608, 72267300476, 579336907254, 4822070246225, 41597773001612, 371306237988959, 3424303740576440, 32583334570211654, 319487530199710232, 3224337031346853361
Offset: 0
Keywords
Programs
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Mathematica
nmax = 25; CoefficientList[Series[Sin[x] Exp[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]! Table[Sum[(-1)^k Binomial[n, 2 k + 1] BellB[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}], {n, 0, 25}]
Formula
a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * Bell(n-2*k-1).
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
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