A346844 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^5 / 5!.
1, 21, 287, 3290, 34671, 350889, 3492511, 34669734, 346231886, 3497726232, 35872743270, 374387203190, 3982122624117, 43207791878715, 478532965417529, 5411213661200830, 62482405993313229, 736696756305382411, 8868148033487285103, 108969560832001750716
Offset: 5
Keywords
Programs
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Maple
b:= proc(n, m) option remember; `if`(n=0, binomial(m, 5), m*b(n-1, m)+b(n-1, m+1)) end: a:= n-> b(n, 0): seq(a(n), n=5..24); # Alois P. Heinz, Aug 05 2021 -
Mathematica
nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^5/5!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] & Table[Sum[StirlingS2[n, k] Binomial[k, 5], {k, 0, n}], {n, 5, 24}] Table[Sum[Binomial[n, k] StirlingS2[k, 5] BellB[n - k], {k, 0, n}], {n, 5, 24}] Table[(-BellB[n] + 89*BellB[n+1] - 145*BellB[n+2] + 75*BellB[n+3] - 15*BellB[n+4] + BellB[n+5])/120, {n, 5, 24}] (* Vaclav Kotesovec, Aug 06 2021 *) -
PARI
my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^5/5!)) \\ Michel Marcus, Aug 06 2021
Formula
a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,5).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,5) * Bell(n-k).
a(n) = (-Bell(n) + 89*Bell(n+1) - 145*Bell(n+2) + 75*Bell(n+3) - 15*Bell(n+4) + Bell(n+5))/120. - Vaclav Kotesovec, Aug 06 2021