A366199 Expansion of e.g.f. exp(4*(exp(x) - 1) + 2*x).

1, 6, 40, 292, 2308, 19580, 177044, 1696572, 17148916, 182114972, 2024979604, 23506175868, 284125820724, 3567957972316, 46454893734612, 625979771144764, 8715626185644916, 125200337417147932, 1853095248414187796, 28225529312569364732, 441925530173009732532

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

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

a(n) = exp(-4) * Sum_{k>=0} 4^k * (k+2)^n / k!.

a(0) = 1; a(n) = 2 * a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

Mathematica

nmax = 20; CoefficientList[Series[Exp[4 (Exp[x] - 1) + 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1) + 2*x))) \\ Michel Marcus, Dec 07 2023

Crossrefs

Cf. A005493, A035009, A078945, A355247, A367889, A367937.

Sequence in context: A005037 A081337 A316912 * A373622 A138240 A358598

Adjacent sequences: A366196 A366197 A366198 * A366200 A366201 A366202

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 05 2023

Status

approved