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A349882 Expansion of Sum_{k>=0} k^2 * x^k/(1 - k^2 * x).

%I A349882 #26 Jan 12 2023 06:44:54
%S A349882 0,1,5,26,162,1267,12343,145652,2036148,33192789,622384729,
%T A349882 13263528350,318121600694,8517247764135,252725694989611,
%U A349882 8258153081400856,295515712276222952,11523986940937975401,487562536078882116717,22291094729329088403298
%N A349882 Expansion of Sum_{k>=0} k^2 * x^k/(1 - k^2 * x).
%H A349882 Seiichi Manyama, <a href="/A349882/b349882.txt">Table of n, a(n) for n = 0..339</a>
%F A349882 a(n) = Sum_{k=0..n} k^(2*(n-k+1)).
%F A349882 a(n) = A234568(n+1) - 1. - _Hugo Pfoertner_, Dec 04 2021
%F A349882 a(n) ~ sqrt(Pi) * ((n+1)/LambertW(exp(1)*(n+1)))^(5/2 + 2*n - 2*(n+1)/LambertW(exp(1)*(n+1))) / sqrt(1 + LambertW(exp(1)*(n+1))). - _Vaclav Kotesovec_, Dec 04 2021
%F A349882 G.f.: Sum_{k>=1} x^k/(1 - (k+1)^2 * x). - _Seiichi Manyama_, Jan 12 2023
%t A349882 a[n_] := Sum[If[k == n - k + 1 == 0, 1, k^(2*(n - k + 1))], {k, 0, n}]; Array[a, 20, 0] (* _Amiram Eldar_, Dec 04 2021 *)
%o A349882 (PARI) a(n, s=2, t=2) = sum(k=0, n, k^(t*(n-k)+s));
%o A349882 (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^2*x^k/(1-k^2*x))))
%o A349882 (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k/(1-(k+1)^2*x)))) \\ _Seiichi Manyama_, Jan 12 2023
%Y A349882 Cf. A003101, A234568, A249459, A349863, A359659.
%K A349882 nonn
%O A349882 0,3
%A A349882 _Seiichi Manyama_, Dec 03 2021