A346843 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^4 / 4!.
1, 15, 155, 1400, 11991, 101031, 853315, 7300260, 63641006, 567304452, 5181338526, 48538121450, 466611951261, 4603782469653, 46613101232933, 484188586821376, 5157850655391981, 56321812548867229, 630125374420189131, 7219368394888423554, 84658119388335562972
Offset: 4
Keywords
Programs
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Maple
b:= proc(n, m) option remember; `if`(n=0, binomial(m, 4), m*b(n-1, m)+b(n-1, m+1)) end: a:= n-> b(n, 0): seq(a(n), n=4..24); # Alois P. Heinz, Aug 05 2021 -
Mathematica
nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^4/4!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] & Table[Sum[StirlingS2[n, k] Binomial[k, 4], {k, 0, n}], {n, 4, 24}] Table[Sum[Binomial[n, k] StirlingS2[k, 4] BellB[n - k], {k, 0, n}], {n, 4, 24}] Table[(BellB[n] - 24*BellB[n+1] + 29*BellB[n+2] - 10*BellB[n+3] + BellB[n+4])/24, {n, 4, 24}] (* Vaclav Kotesovec, Aug 06 2021 *) With[{nn=30},Drop[CoefficientList[Series[(Exp[Exp[x]-1](Exp[x]-1)^4)/4!,{x,0,nn}],x] Range[0,nn]!,4]] (* Harvey P. Dale, Oct 03 2024 *) -
PARI
my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^4/4!)) \\ Michel Marcus, Aug 06 2021
Formula
a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,4).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,4) * Bell(n-k).
a(n) = (Bell(n) - 24*Bell(n+1) + 29*Bell(n+2) - 10*Bell(n+3) + Bell(n+4))/24. - Vaclav Kotesovec, Aug 06 2021