A347339 E.g.f.: exp( (exp(x) - 1) * exp(exp(x) - 1) ).

1, 1, 4, 20, 123, 902, 7656, 73509, 785154, 9213324, 117624569, 1621028312, 23959376436, 377730250003, 6322478398476, 111904530008040, 2087093471665987, 40891426070289970, 839329531471890724, 18004595602417946685, 402747680140030433886, 9376084240910510840672, 226760664399026618376569

Offset: 0

Author

Ilya Gutkovskiy, Aug 27 2021

Keywords

• nonn

Comments

Exponential transform of A138378.

Stirling transform of A000248.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..465

Crossrefs

Cf. A000110, A000248, A000258, A005493, A005727, A087761, A138378.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*j*binomial(n-1, j-1), 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, Aug 27 2021

Mathematica

nmax = 22; CoefficientList[Series[Exp[(Exp[x] - 1) Exp[Exp[x] - 1]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (BellB[k + 1] - BellB[k]) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

PARI

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

Formula

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

a(n) = Sum_{k=0..n} Stirling2(n,k) * A000248(k).