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

%I A353204 #27 Feb 23 2023 18:07:23
%S A353204 1,-1,-1,2,1,9,1,146,-167,5363,-16109,355354,-1844831,37142117,
%T A353204 -279336147,5615638874,-55537087439,1157104637831,-14174652825017,
%U A353204 311217052899986,-4538156701549279,105770066665097729,-1785320722016719271,44287095132343348482
%N A353204 Expansion of e.g.f. (1 - x^2)^(1 + 1/x).
%H A353204 Robert Israel, <a href="/A353204/b353204.txt">Table of n, a(n) for n = 0..450</a>
%F A353204 E.g.f.: exp( -Sum{k >= 1} x^k/A110654(k) ).
%F A353204 a(0) = 1; a(n) = -(n-1)! * Sum_{k=1..n} k/A110654(k) * a(n-k)/(n-k)!.
%F A353204 a(n) ~ -(-1)^n * n! / n^2 * (1 - 2*log(n)/n). - _Vaclav Kotesovec_, May 09 2022
%p A353204 S:=series((1-x^2)^(1+1/x),x,31):
%p A353204 seq(coeff(S,x,i)*i!,i=0..30); # _Robert Israel_, Nov 01 2022
%t A353204 nmax = 25; CoefficientList[Series[(1 - x^2)^(1 + 1/x), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, May 09 2022 *)
%o A353204 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^2)^(1+1/x)))
%o A353204 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, x^k/((k+1)\2)))))
%o A353204 (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;
%Y A353204 Cf. A110654, A193281, A353205.
%K A353204 sign
%O A353204 0,4
%A A353204 _Seiichi Manyama_, Apr 30 2022