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 A386514 #28 Aug 28 2025 00:26:52
%S A386514 1,0,2,18,156,1560,18480,254520,3973200,68947200,1312748640,
%T A386514 27175024800,607314818880,14566195163520,373027570755840,
%U A386514 10154293067318400,292659790712889600,8899747730037964800,284685195814757337600,9553060139009702515200,335468448755976164428800
%N A386514 Expansion of e.g.f. exp(x^2/(1-x)^3).
%C A386514 For n > 0, a(n) is the number of ways to linearly order n distinguishable objects into one or several lines and then choose 2 objects from each line. If the lines are also linearly ordered see A364524.
%C A386514 A001804(n) is the number of ways if only 1 line is used.
%F A386514 From _Vaclav Kotesovec_, Aug 24 2025: (Start)
%F A386514 For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n+k-1, 3*k-1) / k!.
%F A386514 a(n) = 4*(n-1)*a(n-1) - 2*(n-1)*(3*n-7)*a(n-2) + (n-2)*(n-1)*(4*n-11)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
%F A386514 a(n) ~ 3^(1/8) * exp(1/27 - 3^(-5/4)*n^(1/4)/8 - 3^(-1/2)*n^(1/2)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n-1/8) / 2. (End)
%e A386514 a(6)=18480 since there are 10800 ways using one line, 4320 ways with 2 lines using 2 and 4 objects, 3240 ways with 2 lines of 3 objects each, and 120 ways with 3 lines of 2 objects each.
%t A386514 nmax = 20; CoefficientList[Series[E^(x^2/(1-x)^3), {x, 0, nmax}], x] * Range[0, nmax]! (* or *)
%t A386514 nmax = 20; Join[{1}, Table[n!*Sum[Binomial[n + k - 1, 3*k - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* _Vaclav Kotesovec_, Aug 24 2025 *)
%Y A386514 Cf. A001804, A364524.
%Y A386514 Cf. A293012, A000262, A082579, A091695, A361283.
%Y A386514 Cf. A052845, A052887, A387244.
%K A386514 nonn,new
%O A386514 0,3
%A A386514 _Enrique Navarrete_, Aug 23 2025