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

%I A346842 #14 Jun 28 2022 07:21:24
%S A346842 1,10,75,520,3556,24626,174805,1279240,9677151,75750752,613656836,
%T A346842 5142797660,44557627661,398786697398,3683575764083,35084121263136,
%U A346842 344242894197456,3476490965903174,36104281709286841,385257741260565844,4220537246457019687,47432055430482106880
%N A346842 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^3 / 3!.
%F A346842 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,3).
%F A346842 a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,3) * Bell(n-k).
%F A346842 a(n) = (Bell(n+3) - 6*Bell(n+2) + 8*Bell(n+1) - Bell(n))/6. - _Vaclav Kotesovec_, Aug 06 2021
%F A346842 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
%p A346842 b:= proc(n, m) option remember;
%p A346842      `if`(n=0, binomial(m, 3), m*b(n-1, m)+b(n-1, m+1))
%p A346842     end:
%p A346842 a:= n-> b(n, 0):
%p A346842 seq(a(n), n=3..24);  # _Alois P. Heinz_, Aug 05 2021
%t A346842 nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^3/3!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
%t A346842 Table[Sum[StirlingS2[n, k] Binomial[k, 3], {k, 0, n}], {n, 3, 24}]
%t A346842 Table[Sum[Binomial[n, k] StirlingS2[k, 3] BellB[n - k], {k, 0, n}], {n, 3, 24}]
%t A346842 Table[(BellB[n+3] - 6*BellB[n+2] + 8*BellB[n+1] - BellB[n])/6, {n, 3, 24}] (* _Vaclav Kotesovec_, Aug 06 2021 *)
%o A346842 (PARI) my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^3/3!)) \\ _Michel Marcus_, Aug 06 2021
%Y A346842 Cf. A000110, A000292, A000392, A003128, A005493, A049020, A189924, A346843, A346844.
%K A346842 nonn
%O A346842 3,2
%A A346842 _Ilya Gutkovskiy_, Aug 05 2021