A308862 Expansion of e.g.f. 1/(1 - x*(1 + 3*x + x^2)*exp(x)).

1, 1, 10, 81, 976, 14505, 258456, 5377897, 127852096, 3419620209, 101625743080, 3322169384721, 118475520287136, 4577175039397753, 190436902905933880, 8489222610046324665, 403657900923994965376, 20393319895130130117729, 1090902632352025316904648

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..381

Formula

E.g.f.: 1 / (1 - Sum_{k>=1} k^3*x^k/k!).

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

a(n) ~ n! / (r^(n+1) * exp(r) * (1 + 7*r + 6*r^2 + r^3)), where r = 0.33649177041401456061485914122406146158245451810028937972189... is the root of the equation exp(r)*r*(1 + 3*r + r^2) = 1. - Vaclav Kotesovec, Jun 29 2019

Mathematica

nmax = 18; CoefficientList[Series[1/(1 - x (1 + 3 x + x^2) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

Prog

(PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1 - x*(1 + 3*x + x^2)*exp(x)))) \\ Michel Marcus, Mar 10 2022

Crossrefs

Cf. A000578, A006153, A144109, A279358, A302870, A308861.

Sequence in context: A363559 A350503 A277205 * A193327 A043458 A038487

Adjacent sequences: A308859 A308860 A308861 * A308863 A308864 A308865

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 29 2019

Status

approved