A347340 E.g.f.: exp( exp(exp(x) - 1) - exp(x) ).

1, 0, 1, 4, 17, 91, 587, 4327, 35604, 323316, 3210600, 34574453, 400893066, 4975247460, 65755573847, 921535225267, 13643496840808, 212688569520955, 3480978391442106, 59657975022473437, 1068151956803180295, 19937983367649562025, 387243759600707804811, 7812456801157894913964

Offset: 0

Author

Ilya Gutkovskiy, Aug 27 2021

Keywords

• nonn

Comments

Exponential transform of A058692.

Stirling transform of A000296.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..480

Crossrefs

Cf. A000110, A000166, A000258, A000296, A000587, A052852, A058692, A182386, A288268.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*binomial(n-1, j-1), j=2..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..23);  # Alois P. Heinz, Aug 27 2021
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*
      `if`(t=0, 1, b(j, 0)-1)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 02 2021

Mathematica

nmax = 23; CoefficientList[Series[Exp[Exp[Exp[x] - 1] - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (BellB[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

PARI

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(exp(x)-1)-exp(x)))) \\ Michel Marcus, Aug 27 2021

Formula

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

a(n) = Sum_{k=0..n} Stirling2(n,k) * A000296(k).

a(n) = Sum_{k=0..n} binomial(n,k) * A000258(k) * A000587(n-k).