A352617 - OEIS

%I #14 Mar 25 2022 10:14:18

%S 1,2,5,16,60,254,1199,6206,34827,210264,1355992,9288954,67279309,

%T 513149498,4107383185,34398823888,300629113292,2735356900806,

%U 25857446103571,253472859754918,2572266378189583,26981781750668760,292136508070103208,3260640536587635410,37472102225288489529

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

%H Seiichi Manyama, <a href="/A352617/b352617.txt">Table of n, a(n) for n = 0..565</a>

%F a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} binomial(n-1,k-1) * (3 - (-1)^k) * a(n-k).

%F a(n) = Sum_{k=0..n} binomial(n,k) * A000110(k) * A003724(n-k).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A005046(k) * A352279(n-2*k).

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

%p       a(n-k)*binomial(n-1, k-1)*(1+(k mod 2)), k=1..n))

%p     end:

%p seq(a(n), n=0..24);  # _Alois P. Heinz_, Mar 24 2022

%t nmax = 24; CoefficientList[Series[Exp[Exp[x] + Sinh[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!

%t 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 (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp( exp(x) + sinh(x) - 1 ))) \\ _Michel Marcus_, Mar 24 2022

%Y Cf. A000110, A001861, A003724, A005046, A352279, A352326.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Mar 24 2022