A346395 Expansion of e.g.f. -log(1 - x) * exp(3*x).

0, 1, 7, 38, 192, 969, 5115, 29322, 187992, 1370745, 11392839, 107043606, 1122823944, 12989320785, 164040593067, 2243143392138, 32994768719376, 519229765892241, 8701862242296807, 154700700117472422, 2907409255935736752, 57588370882960384377, 1198954118077558162875

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, A053486, A346394, A346396.

Programs

Mathematica

nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[3 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[3^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+2)*v[i]-3*(i-1)*v[i-1]+3^(i-1)); v; \\ Seiichi Manyama, May 27 2022

Formula

a(n) = n! * Sum_{k=0..n-1} 3^k / ((n-k) * k!).

a(n) ~ exp(3) * (n-1)!. - Vaclav Kotesovec, Aug 09 2021

a(0) = 0, a(1) = 1, a(n) = (n+2) * a(n-1) - 3 * (n-1) * a(n-2) + 3^(n-1). - Seiichi Manyama, May 27 2022