%I #12 Sep 03 2021 12:30:39

%S 1,0,1,4,17,91,587,4327,35604,323316,3210600,34574453,400893066,

%T 4975247460,65755573847,921535225267,13643496840808,212688569520955,

%U 3480978391442106,59657975022473437,1068151956803180295,19937983367649562025,387243759600707804811,7812456801157894913964

%N E.g.f.: exp( exp(exp(x) - 1) - exp(x) ).

%C Exponential transform of A058692.

%C Stirling transform of A000296.

%H Alois P. Heinz, <a href="/A347340/b347340.txt">Table of n, a(n) for n = 0..480</a>

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * (Bell(k) - 1) * a(n-k).

%F a(n) = Sum_{k=0..n} Stirling2(n,k) * A000296(k).

%F a(n) = Sum_{k=0..n} binomial(n,k) * A000258(k) * A000587(n-k).

%p g:= proc(n) option remember; `if`(n=0, 1,

%p add(g(n-j)*binomial(n-1, j-1), j=2..n))

%p end:

%p b:= proc(n, m) option remember; `if`(n=0,

%p g(m), m*b(n-1, m)+b(n-1, m+1))

%p end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..23); # _Alois P. Heinz_, Aug 27 2021

%p # second Maple program:

%p b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*

%p `if`(t=0, 1, b(j, 0)-1)*binomial(n-1, j-1), j=1..n))

%p end:

%p a:= n-> b(n, 1):

%p seq(a(n), n=0..23); # _Alois P. Heinz_, Sep 02 2021

%t nmax = 23; CoefficientList[Series[Exp[Exp[Exp[x] - 1] - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (BellB[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(exp(exp(exp(x)-1)-exp(x)))) \\ _Michel Marcus_, Aug 27 2021

%Y Cf. A000110, A000166, A000258, A000296, A000587, A052852, A058692, A182386, A288268.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Aug 27 2021