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A352617 Expansion of e.g.f. exp( exp(x) + sinh(x) - 1 ).

%I A352617 #14 Mar 25 2022 10:14:18
%S A352617 1,2,5,16,60,254,1199,6206,34827,210264,1355992,9288954,67279309,
%T A352617 513149498,4107383185,34398823888,300629113292,2735356900806,
%U A352617 25857446103571,253472859754918,2572266378189583,26981781750668760,292136508070103208,3260640536587635410,37472102225288489529
%N A352617 Expansion of e.g.f. exp( exp(x) + sinh(x) - 1 ).
%H A352617 Seiichi Manyama, <a href="/A352617/b352617.txt">Table of n, a(n) for n = 0..565</a>
%F A352617 a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} binomial(n-1,k-1) * (3 - (-1)^k) * a(n-k).
%F A352617 a(n) = Sum_{k=0..n} binomial(n,k) * A000110(k) * A003724(n-k).
%F A352617 a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A005046(k) * A352279(n-2*k).
%p A352617 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A352617       a(n-k)*binomial(n-1, k-1)*(1+(k mod 2)), k=1..n))
%p A352617     end:
%p A352617 seq(a(n), n=0..24);  # _Alois P. Heinz_, Mar 24 2022
%t A352617 nmax = 24; CoefficientList[Series[Exp[Exp[x] + Sinh[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t A352617 a[0] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n - 1, k - 1] (3 - (-1)^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
%o A352617 (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp( exp(x) + sinh(x) - 1 ))) \\ _Michel Marcus_, Mar 24 2022
%Y A352617 Cf. A000110, A001861, A003724, A005046, A352279, A352326.
%K A352617 nonn
%O A352617 0,2
%A A352617 _Ilya Gutkovskiy_, Mar 24 2022