%I #12 Aug 06 2021 13:58:12
%S 1,2,10,66,538,5186,57402,714594,9853978,148774914,2436823034,
%T 42979319202,811254807770,16302732719682,347248840767162,
%U 7809649226242530,184831773033020826,4589793199157616770,119272846472231229818,3235960069037751550498,91466308730323104617050
%N E.g.f.: exp(exp(2*(exp(x) - 1)) - 1).
%H Alois P. Heinz, <a href="/A346417/b346417.txt">Table of n, a(n) for n = 0..449</a>
%F a(n) = Sum_{k=0..n} Stirling2(n,k) * 2^k * Bell(k).
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001861(k) * a(n-k).
%p b:= proc(n, t, m) option remember; `if`(n=0, `if`(t=1, 1,
%p b(m, 1, 0)*2^m) , m*b(n-1, t, m)+b(n-1, t, m+1))
%p end:
%p a:= n-> b(n, 0$2):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Aug 06 2021
%t nmax = 20; CoefficientList[Series[Exp[Exp[2 (Exp[x] - 1)] - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[StirlingS2[n, k] 2^k BellB[k], {k, 0, n}], {n, 0, 20}]
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] BellB[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(exp(exp(2*(exp(x) - 1)) - 1))) \\ _Michel Marcus_, Jul 19 2021
%Y Cf. A000258, A000898, A001861, A055882, A126390, A136658.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jul 16 2021
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