A346394 Expansion of e.g.f. -log(1 - x) * exp(2*x).

0, 1, 5, 20, 78, 324, 1520, 8336, 53872, 405600, 3492416, 33798016, 362543104, 4264455168, 54540715008, 753246711808, 11168972683264, 176937613586432, 2982069587042304, 53271637651996672, 1005385746384846848, 19987620914387812352, 417489079682758213632

Offset: 0

Author

Ilya Gutkovskiy, Jul 15 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A002104, A010842, A066534, A346395, A346396.

Programs

Mathematica

nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[2 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[2^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 22}]

PARI

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i+1)*v[i]-2*(i-1)*v[i-1]+2^(i-1)); v; \\ Seiichi Manyama, May 27 2022

Formula

a(n) = n! * Sum_{k=0..n-1} 2^k / ((n-k) * k!).

a(n) = Sum_{k=0..n} binomial(n,k) * A002104(k).

a(n) ~ exp(2) * (n-1)!. - Vaclav Kotesovec, Aug 09 2021

a(0) = 0, a(1) = 1, a(n) = (n+1) * a(n-1) - 2 * (n-1) * a(n-2) + 2^(n-1). - Seiichi Manyama, May 27 2022