A346397 Expansion of e.g.f. -log(1 - x) * exp(-2*x).

0, 1, -3, 8, -18, 44, -80, 272, 112, 5280, 38464, 414336, 4573184, 55680000, 731374592, 10335551488, 156303374336, 2518984953856, 43099088904192, 780268881068032, 14902336355991552, 299452809651617792, 6315501510330286080, 139485953831281098752, 3219718099932087844864

Offset: 0

Author

Ilya Gutkovskiy, Jul 15 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..451

Crossrefs

Cf. A000023, A002741, A335111, A346398.

Programs

Mathematica

nmax = 24; 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, 24}]

PARI

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i-3)*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} (-1)^(n-k) * binomial(n,k) * A002741(k).

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

a(n) ~ exp(-2) * (n-1)!. - Vaclav Kotesovec, Jun 08 2022