A367890 Expansion of e.g.f. exp(3*(exp(x) - 1 - x)).

1, 0, 3, 3, 30, 93, 633, 3342, 22809, 156063, 1183872, 9453711, 80455125, 721576560, 6809391111, 67332650007, 695777512638, 7493572404345, 83926492573341, 975467527353750, 11744536832206149, 146234590864310019, 1880198749437144456, 24928860500681953683

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Formula

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

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

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

a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A027710(k).

Mathematica

nmax = 23; CoefficientList[Series[Exp[3 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 23}]

Prog

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

Crossrefs

Cf. A000296, A027710, A194689, A346738, A355253, A355254, A367888, A367891.

Sequence in context: A351990 A151480 A096351 * A344934 A086667 A067098

Adjacent sequences: A367887 A367888 A367889 * A367891 A367892 A367893

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 04 2023

Status

approved