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

%I #10 Dec 29 2023 06:24:42

%S 1,2,7,38,272,2444,26306,330588,4746360,76675584,1376187072,

%T 27171073632,585216675600,13655030234208,343124183767920,

%U 9237920561327904,265292717180631552,8094790891854169344,261522698597072168832,8918551194519088836864

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

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

%F a(n) ~ n! * LambertW(exp(2)) / ((LambertW(exp(2)) + 1) * (LambertW(exp(2)) - 1)^(n+1)) . - _Vaclav Kotesovec_, Dec 29 2023

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

%Y Cf. A006252, A368233.

%Y Cf. A338448.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 18 2023