A328182 Expansion of e.g.f. 1 / (2 - exp(3*x)).

1, 3, 27, 351, 6075, 131463, 3413907, 103429791, 3581223435, 139498558263, 6037616347587, 287444492409231, 14929010774254395, 839982382565841063, 50897213545996785267, 3304312091004451756671, 228821504027595115886955, 16836102104577636004291863, 1311625494765417347634022947

Offset: 0

Author

Ilya Gutkovskiy, Oct 06 2019

Keywords

• nonn

Crossrefs

Cf. A000670, A216794, A247452, A328183.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*3^j, j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 06 2019

Mathematica

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

Formula

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

a(n) = Sum_{k>=0} (3*k)^n / 2^(k + 1).

a(n) = 3^n * A000670(n).

a(n) ~ n! * 3^n / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 09 2021