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Expansion of e.g.f. 1/(exp(x) - log(1 + x)).
4

%I #18 Mar 07 2022 02:09:57

%S 1,0,-2,1,17,-17,-401,817,16197,-49861,-1123633,5354787,105696447,

%T -682603651,-14697824519,131535803133,2457119246745,-28321054685609,

%U -572811846560453,8626026427105983,146289547341006011,-2784279036040263575,-51756654994427512331

%N Expansion of e.g.f. 1/(exp(x) - log(1 + x)).

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

%t m = 22; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - Log[1 + x]), {x, 0, m}], x] (* _Amiram Eldar_, Mar 06 2022 *)

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-log(1+x))))

%o (PARI) a(n) = if(n==0, 1, -sum(k=1, n, ((-1)^k*(k-1)!+1)*binomial(n, k)*a(n-k)));

%Y Cf. A352139, A352146, A352147.

%Y Cf. A235378.

%K sign

%O 0,3

%A _Seiichi Manyama_, Mar 06 2022