A352294 -id:A352294 - cp's OEIS frontend

A352295 Expansion of e.g.f. 1/(exp(x) - x/(1 + x)).

1, 0, -3, 5, 29, -181, -401, 9645, -14183, -689257, 4826171, 55700633, -1024570955, -2770525005, 221566919911, -1028838834811, -49439771820367, 723165789334703, 9903852025111027, -362150510124039471, -463774017017434739, 169793689786411161995

Offset: 0

Seiichi Manyama, Mar 11 2022

sign

Cf. A352138, A352293, A352294.

m = 21; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x/(1 + x)), {x, 0, m}], x] (* Amiram Eldar, Mar 11 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x/(1+x))))

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

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

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

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jul 14 2022

nonn

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

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

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

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

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;

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

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A352295 Expansion of e.g.f. 1/(exp(x) - x/(1 + x)).

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

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