A353204 Expansion of e.g.f. (1 - x^2)^(1 + 1/x).

1, -1, -1, 2, 1, 9, 1, 146, -167, 5363, -16109, 355354, -1844831, 37142117, -279336147, 5615638874, -55537087439, 1157104637831, -14174652825017, 311217052899986, -4538156701549279, 105770066665097729, -1785320722016719271, 44287095132343348482

Offset: 0

Author

Seiichi Manyama, Apr 30 2022

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A110654, A193281, A353205.

Programs

Maple

S:=series((1-x^2)^(1+1/x),x,31):
seq(coeff(S,x,i)*i!,i=0..30); # Robert Israel, Nov 01 2022

Mathematica

nmax = 25; CoefficientList[Series[(1 - x^2)^(1 + 1/x), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 09 2022 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^2)^(1+1/x)))

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, x^k/((k+1)\2)))))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!*sum(j=1, i, j/((j+1)\2)*v[i-j+1]/(i-j)!)); v;

Formula

E.g.f.: exp( -Sum{k >= 1} x^k/A110654(k) ).

a(0) = 1; a(n) = -(n-1)! * Sum_{k=1..n} k/A110654(k) * a(n-k)/(n-k)!.

a(n) ~ -(-1)^n * n! / n^2 * (1 - 2*log(n)/n). - Vaclav Kotesovec, May 09 2022