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 A335345 #17 Jun 17 2024 08:50:08
%S A335345 1,0,1,9,75,690,7305,89145,1237425,19221300,329371245,6157738125,
%T A335345 124551652995,2707913238030,62945320162725,1557291398788125,
%U A335345 40844991621859425,1131753403094113800,33025920511859300025,1012128709342410284625,32494107983067177522075
%N A335345 Expansion of e.g.f. exp(x^2/(2*(1 - x)^3)).
%F A335345 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001809(k) * a(n-k).
%F A335345 D-finite with recurrence 2*a(n) +8*(-n+1)*a(n-1) +2*(n-1)*(6*n-13)*a(n-2) -(n-1)*(n-2)*(8*n-23)*a(n-3) +2*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jun 05 2020
%F A335345 a(n) ~ 2^(-9/8) * 3^(1/8) * n^(n - 1/8) * exp(1/54 - n^(1/4)/(2^(15/4)*3^(5/4)) - sqrt(6*n)/12 + 2^(7/4)*3^(-3/4)*n^(3/4) - n). - _Vaclav Kotesovec_, Jun 11 2020
%F A335345 a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n+k-1,n-2*k)/(2^k * k!). - _Seiichi Manyama_, Jun 17 2024
%t A335345 nmax = 20; CoefficientList[Series[Exp[x^2/(2 (1 - x)^3)], {x, 0, nmax}], x] Range[0, nmax]!
%t A335345 a[0] = 1; a[n_] := a[n] = (1/4) Sum[Binomial[n - 1, k - 1] k (k - 1) k! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
%o A335345 (PARI) seq(n)=Vec(serlaplace(exp(x^2/(2*(1 - x)^3) + O(x*x^n)))) \\ _Andrew Howroyd_, Jun 02 2020
%Y A335345 Cf. A001809, A091695, A185369, A335344.
%K A335345 nonn
%O A335345 0,4
%A A335345 _Ilya Gutkovskiy_, Jun 02 2020