A346432 a(0) = 1; a(n) = n! * Sum_{k=0..n-1} (n-k+1) * a(k) / k!.

1, 2, 14, 144, 1968, 33600, 688320, 16450560, 449326080, 13806858240, 471395635200, 17703899136000, 725338710835200, 32193996432998400, 1538840509503897600, 78808952068374528000, 4305129487814098944000, 249876735246162984960000, 15356385691181506363392000

Offset: 0

Author

Ilya Gutkovskiy, Jul 17 2021

Keywords

• nonn

Crossrefs

Cf. A000670, A001339, A002866, A003480, A007840, A052555, A052567, A136658, A216794, A308939, A346433.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = n! Sum[(n - k + 1) a[k]/k!, {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 - 1/(1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) StirlingS1[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]

PARI

my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - 1 / (1 - x)^2))) \\ Michel Marcus, Jul 18 2021

Formula

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

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

a(0) = 1, a(1) = 2, a(2) = 14; a(n) = 4 * n * a(n-1) - 2 * n * (n-1) * a(n-2).

a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n,k) * 2^k * A000670(k).

a(n) = n! * A003480(n).