A352624 Expansion of e.g.f. exp(exp(x) + cosh(x) - 2).

1, 1, 3, 8, 31, 122, 579, 2886, 16139, 95358, 611111, 4128830, 29709695, 224400022, 1785322699, 14841968646, 129015458195, 1167021383902, 10979895178511, 107113768171950, 1082508179141031, 11308614423992102, 121995294474174963, 1356835055606851286, 15542964081299602811

Offset: 0

Author

Ilya Gutkovskiy, Mar 24 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..566

Crossrefs

Cf. A000110, A000807, A001861, A003724, A005046, A352327, A352617.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n-1, k-1)*(2-(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] + Cosh[x] - 2], {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}]

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..floor(n/2)} binomial(n,2*k) * A005046(k) * A000110(n-2*k).

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A000807(k) * A003724(n-2*k).