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 A360816 #17 Aug 04 2025 09:46:25
%S A360816 1,0,1,2,19,100,1118,10034,134993,1715140,27589661,449763360,
%T A360816 8522965956,168431719308,3698624353289,85523954588806,
%U A360816 2142927489388319,56618555339223572,1596938935380604858,47399670488829289678,1487559109670284821841
%N A360816 Expansion of Sum_{k>=0} (k*x)^(2*k) / (1 - k*x)^(k+1).
%H A360816 Seiichi Manyama, <a href="/A360816/b360816.txt">Table of n, a(n) for n = 0..418</a>
%F A360816 a(n) = Sum_{k=0..floor(n/2)} k^n * binomial(n-k,k).
%F A360816 a(n) ~ (1-r)^(1 + n*(1-r)) * r^(1/2 + n*(1-r)) * n^n / (sqrt(1 - 2*r + 2*r^2) * (1-2*r)^(n*(1-2*r))), where r = 0.42401262950134202779147542659633991972637211375... is the root of the equation log(r*(1-r)) - 2*log(1-2*r) = 1/r. - _Vaclav Kotesovec_, Aug 04 2025
%t A360816 Join[{1}, Table[Sum[k^n * Binomial[n-k,k], {k,0,n/2}], {n,1,20}]] (* _Vaclav Kotesovec_, Aug 04 2025 *)
%o A360816 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^(2*k)/(1-k*x)^(k+1)))
%o A360816 (PARI) a(n) = sum(k=0, n\2, k^n*binomial(n-k, k));
%Y A360816 Cf. A072034, A360817.
%Y A360816 Cf. A360814.
%K A360816 nonn
%O A360816 0,4
%A A360816 _Seiichi Manyama_, Feb 21 2023