A346844 -id:A346844 - cp's OEIS frontend

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

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000110, A000292, A000392, A003128, A005493, A049020, A189924, A346843, A346844.

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

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 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^3/3!)) \\ Michel Marcus, Aug 06 2021

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

A346843 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^4 / 4!.

Original entry on oeis.org

1, 15, 155, 1400, 11991, 101031, 853315, 7300260, 63641006, 567304452, 5181338526, 48538121450, 466611951261, 4603782469653, 46613101232933, 484188586821376, 5157850655391981, 56321812548867229, 630125374420189131, 7219368394888423554, 84658119388335562972

Offset: 4

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000110, A000332, A000453, A003128, A005493, A049020, A346842, A346844.

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

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 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^4/4!)) \\ Michel Marcus, Aug 06 2021

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

cp's OEIS Frontend

A346842 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^3 / 3!.

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