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

1, 2, 5, 16, 60, 254, 1199, 6206, 34827, 210264, 1355992, 9288954, 67279309, 513149498, 4107383185, 34398823888, 300629113292, 2735356900806, 25857446103571, 253472859754918, 2572266378189583, 26981781750668760, 292136508070103208, 3260640536587635410, 37472102225288489529

Offset: 0

Author

Ilya Gutkovskiy, Mar 24 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..565

Crossrefs

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

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n-1, k-1)*(1+(k mod 2)), k=1..n))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Mar 24 2022

Mathematica

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

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp( exp(x) + sinh(x) - 1 ))) \\ Michel Marcus, Mar 24 2022

Formula

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

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

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