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A330021 Expansion of e.g.f. exp(sinh(exp(x) - 1)).

%I A330021 #9 Jun 23 2023 18:04:13
%S A330021 1,1,2,6,25,128,754,5001,37048,303930,2732395,26657106,280039786,
%T A330021 3149224991,37729906686,479570263690,6442902231289,91186621152460,
%U A330021 1355582225366134,21112253012491481,343672026658191836,5834977672879651390,103130592695715620419
%N A330021 Expansion of e.g.f. exp(sinh(exp(x) - 1)).
%C A330021 Stirling transform of A003724.
%C A330021 Exponential transform of A024429.
%H A330021 Alois P. Heinz, <a href="/A330021/b330021.txt">Table of n, a(n) for n = 0..485</a>
%F A330021 a(n) = Sum_{k=0..n} Stirling2(n,k) * A003724(k).
%F A330021 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A024429(k) * a(n-k).
%p A330021 g:= proc(n) option remember; `if`(n=0, 1, add(
%p A330021       binomial(n-1, j-1)*irem(j, 2)*g(n-j), j=1..n))
%p A330021     end:
%p A330021 b:= proc(n, m) option remember; `if`(n=0,
%p A330021       g(m), m*b(n-1, m)+b(n-1, m+1))
%p A330021     end:
%p A330021 a:= n-> b(n, 0):
%p A330021 seq(a(n), n=0..22);  # _Alois P. Heinz_, Jun 23 2023
%t A330021 nmax = 22; CoefficientList[Series[Exp[Sinh[Exp[x] - 1]], {x, 0, nmax}], x] Range[0, nmax]!
%Y A330021 Cf. A003724, A009218, A011800, A024429, A080831.
%K A330021 nonn
%O A330021 0,3
%A A330021 _Ilya Gutkovskiy_, Nov 27 2019