A355672 Expansion of e.g.f. exp(1/(1-x) - exp(x)).

1, 0, 1, 5, 26, 169, 1329, 12088, 124221, 1422307, 17947550, 247318851, 3693469273, 59396067080, 1022975862713, 18781241965081, 366070181352802, 7547972562003093, 164113696105503057, 3752143293971556144, 89976991297720804061, 2257905394760969948079

Offset: 0

Author

Seiichi Manyama, Jul 14 2022

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Crossrefs

Cf. A000262, A000587, A033312, A186755, A352294.

Programs

Mathematica

nmax = 20; CoefficientList[Series[E^(1/(1-x) - E^x), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 21 2022 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(1/(1-x)-exp(x))))

PARI

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

Formula

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

a(n) ~ exp(1/2 - exp(1) + 2*sqrt(n) - n) * n^(n - 1/4) / sqrt(2). - Vaclav Kotesovec, Jul 21 2022