A337000 E.g.f.: 1 / ((1 - x)*(2 - exp(x))).

1, 2, 7, 34, 211, 1596, 14259, 147106, 1722683, 22591408, 328161643, 5232410646, 90880495347, 1708304787892, 34557610000931, 748647340991818, 17294012137850443, 424368973372593432, 11024176183963527099, 302260934814635426014, 8722906492537092723395

Offset: 0

Author

Ilya Gutkovskiy, Aug 10 2020

Keywords

• nonn

Comments

Exponential convolution of Fubini numbers (A000670) and factorial numbers (A000142).

Crossrefs

Cf. A000142, A000670, A101053.

Programs

Mathematica

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

Formula

a(n) = n! * Sum_{k=0..n} A000670(k) / k!.

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

a(n) ~ n! / (2 * (1 - log(2)) * log(2)^(n+1)). - Vaclav Kotesovec, Aug 09 2021