A346842 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^3 / 3!.
1, 10, 75, 520, 3556, 24626, 174805, 1279240, 9677151, 75750752, 613656836, 5142797660, 44557627661, 398786697398, 3683575764083, 35084121263136, 344242894197456, 3476490965903174, 36104281709286841, 385257741260565844, 4220537246457019687, 47432055430482106880
Offset: 3
Keywords
Programs
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Maple
b:= proc(n, m) option remember; `if`(n=0, binomial(m, 3), m*b(n-1, m)+b(n-1, m+1)) end: a:= n-> b(n, 0): seq(a(n), n=3..24); # Alois P. Heinz, Aug 05 2021 -
Mathematica
nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^3/3!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] & Table[Sum[StirlingS2[n, k] Binomial[k, 3], {k, 0, n}], {n, 3, 24}] Table[Sum[Binomial[n, k] StirlingS2[k, 3] BellB[n - k], {k, 0, n}], {n, 3, 24}] Table[(BellB[n+3] - 6*BellB[n+2] + 8*BellB[n+1] - BellB[n])/6, {n, 3, 24}] (* Vaclav Kotesovec, Aug 06 2021 *) -
PARI
my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^3/3!)) \\ Michel Marcus, Aug 06 2021
Formula
a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,3).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,3) * Bell(n-k).
a(n) = (Bell(n+3) - 6*Bell(n+2) + 8*Bell(n+1) - Bell(n))/6. - Vaclav Kotesovec, Aug 06 2021
a(n) ~ exp(-1 - n + n/LambertW(n)) * (n - LambertW(n))^3 * n^n / (6 * sqrt(1 + LambertW(n)) * LambertW(n)^(n+3)). - Vaclav Kotesovec, Jun 28 2022