A331339 E.g.f.: 1 / (1 + log(1 - x - x^2)).

1, 1, 5, 32, 292, 3294, 44918, 714468, 13002456, 266275200, 6060498672, 151750887936, 4145522908272, 122690391196944, 3910569680464848, 133549150323123744, 4864927063250290176, 188297220693251438208, 7716800776602560577408

Offset: 0

Author

Ilya Gutkovskiy, Jan 14 2020

Keywords

• nonn

Crossrefs

Cf. A000204, A007840, A039647.

Programs

Maple

A331339 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        add(binomial(n,k)*(k-1)!*A000204(k)*procname(n-k),k=1..n) ;
    end if;
end proc:
seq(A331339(n),n=0..42) ; # R. J. Mathar, Aug 20 2021

Mathematica

nmax = 18; CoefficientList[Series[1/(1 + Log[1 - x - x^2]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (k - 1)! LucasL[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k - 1)! * Lucas(k) * a(n-k).

a(n) ~ n! * 2^(n+1) * exp(n/2) / (sqrt(5*exp(1) - 4) * (sqrt(5*exp(1) - 4) - exp(1/2))^(n+1)). - Vaclav Kotesovec, Jan 26 2020