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 A346753 #16 Dec 15 2023 09:02:12
%S A346753 0,0,1,3,9,40,225,1491,11578,102852,1026945,11394955,139091106,
%T A346753 1852061718,26716291693,415033647315,6908006807640,122645325067576,
%U A346753 2313546734841633,46209268921868595,974228913850588750,21620679147700290210,503810188866302511501
%N A346753 Expansion of e.g.f. -log( 1 - x^2 * exp(x) / 2 ).
%F A346753 a(0) = 0; a(n) = binomial(n,2) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,2) * k * a(k).
%F A346753 a(n) ~ (n-1)! / (2*LambertW(1/sqrt(2)))^n. - _Vaclav Kotesovec_, Aug 08 2021
%F A346753 a(n) = n! * Sum_{k=1..floor(n/2)} k^(n-2*k-1)/(2^k * (n-2*k)!). - _Seiichi Manyama_, Dec 14 2023
%t A346753 nmax = 22; CoefficientList[Series[-Log[1 - x^2 Exp[x]/2], {x, 0, nmax}], x] Range[0, nmax]!
%t A346753 a[0] = 0; a[n_] := a[n] = Binomial[n, 2] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]
%Y A346753 Cf. A000217, A009444, A133189, A292953, A305990, A346750, A346754, A346755.
%K A346753 nonn
%O A346753 0,4
%A A346753 _Ilya Gutkovskiy_, Aug 01 2021