A346397 - OEIS

%I #14 Jun 08 2022 10:34:12

%S 0,1,-3,8,-18,44,-80,272,112,5280,38464,414336,4573184,55680000,

%T 731374592,10335551488,156303374336,2518984953856,43099088904192,

%U 780268881068032,14902336355991552,299452809651617792,6315501510330286080,139485953831281098752,3219718099932087844864

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

%H Seiichi Manyama, <a href="/A346397/b346397.txt">Table of n, a(n) for n = 0..451</a>

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

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002741(k).

%F 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

%F a(n) ~ exp(-2) * (n-1)!. - _Vaclav Kotesovec_, Jun 08 2022

%t nmax = 24; CoefficientList[Series[-Log[1 - x] Exp[-2 x], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[n! Sum[(-2)^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 24}]

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=0; v[2]=1; 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

%Y Cf. A000023, A002741, A335111, A346398.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Jul 15 2021