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 A335344 #27 Oct 28 2023 16:32:26
%S A335344 1,0,1,6,39,300,2715,28350,335265,4422600,64298745,1020477150,
%T A335344 17542820295,324552648420,6426708843555,135573281994150,
%U A335344 3034400481137025,71801374285040400,1790523094644709425,46923435009924823350,1289032229351717425575
%N A335344 Expansion of e.g.f. exp(x^2/(2*(1 - x)^2)).
%C A335344 For n>0, a(n) is also the number of ways to split n people into nonempty groups, have each group sit around a circular table, and select 2 people from each table (where two seating arrangements are considered identical if each person has the same left neighbors in both of them). See example below. - _Enrique Navarrete_, Oct 01 2023
%H A335344 Robert Israel, <a href="/A335344/b335344.txt">Table of n, a(n) for n = 0..436</a>
%F A335344 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001286(k) * a(n-k).
%F A335344 D-finite with recurrence a(n) = -(n - 1)*(3*n - 7)*a(n - 2) + 3*(n - 1)*a(n - 1) + (n - 1)*(n - 2)*(n - 3)*a(n - 3). - _Robert Israel_, Jun 04 2020
%F A335344 a(n) ~ n^(n - 1/6) * exp(1/6 - n^(1/3)/2 + 3*n^(2/3)/2 - n) / sqrt(3). - _Vaclav Kotesovec_, Jun 11 2020
%F A335344 a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n-1,n-2*k)/(2^k * k!). - _Seiichi Manyama_, Mar 16 2023
%e A335344 For n = 5, using one table, there are 4! circular seatings and binomial(5,2) ways to select 2 persons, hence 240 ways. Using two tables, the only way we can select 2 persons from each one is seating 3 persons in one table and 2 in the other, which can be done in 20 ways; then choosing 2 persons from each table can be done in 3 ways, hence giving another 60 ways for a total of 300. - _Enrique Navarrete_, Oct 01 2023
%p A335344 f:= gfun:-rectoproc({a(n) = -(n-1)*(3*n-7)*a(n-2) + 3*(n-1)*a(n-1) + (n - 1)*(n - 2)*(n - 3)*a(n-3),a(0)=1,a(1)=0,a(2)=1},a(n),remember):
%p A335344 map(f, [$0..30]); # _Robert Israel_, Jun 04 2020
%t A335344 nmax = 20; CoefficientList[Series[Exp[x^2/(2 (1 - x)^2)], {x, 0, nmax}], x] Range[0, nmax]!
%t A335344 a[0] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n - 1, k - 1] (k - 1) k! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
%o A335344 (PARI) seq(n)=Vec(serlaplace(exp(x^2/(2*(1 - x)^2) + O(x*x^n)))) \\ _Andrew Howroyd_, Jun 02 2020
%Y A335344 Cf. A001286, A052887, A082579, A185369, A335345.
%K A335344 nonn
%O A335344 0,4
%A A335344 _Ilya Gutkovskiy_, Jun 02 2020