A330021 - OEIS

%I #9 Jun 23 2023 18:04:13

%S 1,1,2,6,25,128,754,5001,37048,303930,2732395,26657106,280039786,

%T 3149224991,37729906686,479570263690,6442902231289,91186621152460,

%U 1355582225366134,21112253012491481,343672026658191836,5834977672879651390,103130592695715620419

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

%C Stirling transform of A003724.

%C Exponential transform of A024429.

%H Alois P. Heinz, <a href="/A330021/b330021.txt">Table of n, a(n) for n = 0..485</a>

%F a(n) = Sum_{k=0..n} Stirling2(n,k) * A003724(k).

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A024429(k) * a(n-k).

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

%p       binomial(n-1, j-1)*irem(j, 2)*g(n-j), j=1..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..22);  # _Alois P. Heinz_, Jun 23 2023

%t nmax = 22; CoefficientList[Series[Exp[Sinh[Exp[x] - 1]], {x, 0, nmax}], x] Range[0, nmax]!

%Y Cf. A003724, A009218, A011800, A024429, A080831.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 27 2019