This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349836 #44 Jan 07 2024 06:23:40
%S A349836 1,1,5,44,564,9665,211025,5686104,184813048,7118824417,320295658577,
%T A349836 16626717667348,985178854556524,66005199079345025,4958773228726876257,
%U A349836 414664315430994701616,38344259607889223269168,3898112616839310343827009
%N A349836 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^2 * x).
%H A349836 Seiichi Manyama, <a href="/A349836/b349836.txt">Table of n, a(n) for n = 0..299</a>
%F A349836 a(n) = Sum_{k=0..n} k^(2*n-k).
%F A349836 a(n) ~ sqrt(Pi) * 2^(1 + 2*n - 2*n/LambertW(2*exp(1)*n)) * (n/LambertW(2*exp(1)*n))^(1/2 + 2*n - 2*n/LambertW(2*exp(1)*n)) / sqrt(1 + LambertW(2*exp(1)*n)). - _Vaclav Kotesovec_, Dec 04 2021
%t A349836 Join[{1}, Table[Sum[k^(2*n - k), {k, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Dec 04 2021 *)
%o A349836 (PARI) a(n, t=2) = sum(k=0, n, k^(t*(n-k)+k));
%o A349836 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k^2*x)))
%Y A349836 Cf. A031971, A349883.
%Y A349836 Cf. A249459, A349884, A368505.
%K A349836 nonn
%O A349836 0,3
%A A349836 _Seiichi Manyama_, Dec 03 2021