Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Dec 04 2021 08:54:12
%S 1,1,3,12,76,961,15407,221528,3260936,80774113,2462081967,50963779604,
%T 922244742292,61063845514113,2868669700179871,2019727494212912,
%U -47889136910252848,461395118866593115713,5781219348638565771423,-2108738296748190078596084
%N Expansion of Sum_{k>=0} (k * x)^k/(1 + k^2 * x).
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n-k).
%t a[n_] := Sum[If[k == 2*n - k == 0, 1, (-1)^(n - k) * k^(2*n - k)], {k, 0, n}]; Array[a, 20, 0] (* _Amiram Eldar_, Dec 04 2021 *)
%o (PARI) a(n, t=2) = sum(k=0, n, (-k^t)^(n-k)*k^k);
%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1+k^2*x)))
%Y Cf. A120485, A349885.
%Y Cf. A349856, A349859, A349863.
%K sign
%O 0,3
%A _Seiichi Manyama_, Dec 03 2021