A367938 Expansion of e.g.f. exp(exp(3*x) - 1 - 2*x).

1, 1, 10, 55, 487, 4654, 51463, 632125, 8536492, 125279785, 1981246555, 33530245984, 603797462677, 11513675558701, 231539488842610, 4893151984630579, 108334206855000739, 2505977899186557502, 60419653270442268643, 1515077412621445514089, 39437350309301393464876, 1063746973172416765272589

Offset: 0

Author

Ilya Gutkovskiy, Dec 05 2023

Keywords

• nonn

Crossrefs

Cf. A000110, A124311, A126617, A247452, A284864, A367785, A367888.

Programs

Mathematica

nmax = 21; CoefficientList[Series[Exp[Exp[3 x] - 1 - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Table[Sum[Binomial[n, k] (-2)^(n - k) 3^k BellB[k], {k, 0, n}], {n, 0, 21}]

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(exp(3*x) - 1 - 2*x))) \\ Michel Marcus, Dec 07 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - x * ( 2 * A(x) - 3 * A(x/(1 - 3*x)) / (1 - 3*x) ).

a(n) = exp(-1) * Sum_{k>=0} (3*k-2)^n / k!.

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

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