A367785 Expansion of e.g.f. exp(exp(3*x) - x - 1).

1, 2, 13, 89, 772, 7745, 87949, 1109288, 15332539, 229840361, 3706130914, 63857565095, 1169261937973, 22646779177898, 462143532144937, 9902312863237637, 222119823632283628, 5202170552214520637, 126914730275907871201, 3218552632981994910248, 84686139239808135094879, 2307953474037054591248501

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

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

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

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000110, A000296, A124311, A247452, A284859, A284860, A330605, A367744, A367786.

Sequence in context: A126035 A074617 A337169 * A199489 A300764 A209470

Adjacent sequences: A367782 A367783 A367784 * A367786 A367787 A367788

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 30 2023

Status

approved