A343674 a(0) = 1; a(n) = 4 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

1, 5, 51, 781, 15947, 407021, 12466251, 445452813, 18191122219, 835737327661, 42661645147403, 2395510523568845, 146739531459316587, 9737742346694258157, 695911661109898805323, 53286006304099668950413, 4352120920347139791200171, 377674509364714706139413933, 34702277449656625185428239755

Offset: 0

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

• nonn

Crossrefs

Cf. A000670, A006155, A032032, A045379, A343672, A343673.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 (1 - 2 x) - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1 / (2 * (1 - 2*x) - exp(x)).

a(n) ~ n! * 2^(n-1) / ((1 + LambertW(exp(1/2)/4)) * (1 - 2*LambertW(exp(1/2)/4))^(n+1)). - Vaclav Kotesovec, Jun 20 2022