A367889 Expansion of e.g.f. exp(3*(exp(x) - 1) + 2*x).

1, 5, 28, 173, 1165, 8468, 65923, 546197, 4791214, 44301143, 430158397, 4372004546, 46381674085, 512328076385, 5879362011436, 69958289731457, 861605015493073, 10965899141265500, 144018319806024991, 1949190279770578145, 27153595018237222774

Offset: 0

Author

Ilya Gutkovskiy, Dec 04 2023

Keywords

• nonn

Crossrefs

Cf. A000110, A005493, A035009, A078940, A355247, A367888.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Exp[3 (Exp[x] - 1) + 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + 3 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 2^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 20}]

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) + 2*x))) \\ Michel Marcus, Dec 04 2023

Formula

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

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

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

a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A027710(k).