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

1, 1, 2, 6, 25, 128, 754, 5001, 37048, 303930, 2732395, 26657106, 280039786, 3149224991, 37729906686, 479570263690, 6442902231289, 91186621152460, 1355582225366134, 21112253012491481, 343672026658191836, 5834977672879651390, 103130592695715620419

Offset: 0

Author

Ilya Gutkovskiy, Nov 27 2019

Keywords

• nonn

Comments

Stirling transform of A003724.

Exponential transform of A024429.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..485

Crossrefs

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

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*irem(j, 2)*g(n-j), j=1..n))
    end:
b:= proc(n, m) option remember; `if`(n=0,
      g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..22);  # Alois P. Heinz, Jun 23 2023

Mathematica

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

Formula

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

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