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 A373065 #20 Mar 06 2025 13:59:10
%S A373065 0,0,1,3,18,70,315,1281,5124,19692,73845,270655,974358,3454386,
%T A373065 12090351,41851005,143489160,487862872,1646537193,5520742011,
%U A373065 18402473370,61018727070,201361799331,661617340153,2165293113228,7060738412100,22947399839325,74349575478711,240206320777374,773998144726282
%N A373065 Expansion of e.g.f. (1/2)*(x^2*exp(x))*(cosh(x))^2.
%C A373065 a(n) is the number of ordered set partitions of an n-set into 3 sets such that the first and second sets have an even number of elements, and two elements are selected from the third. "Ordered set partitions", because {}, {1,2}, {(3), (4), 5} is considered to be different from {1,2}, {}, {(3), (4), 5} .
%F A373065 a(n) = binomial(n,2)*(3^(n-2) + (-1)^n + 2)/4.
%e A373065 For n = 5, we have the following cases (allowing empty sets):
%e A373065 {}, {1,2}, {(3), (4), 5} (30 of these),
%e A373065 {1,2}, {}, {(3), (4), 5} (30 of these),
%e A373065 {}, {}, {(1), (2), 3, 4, 5} (10 of these),
%e A373065 where the two elements selected from the third set are in parentheses.
%t A373065 With[{nn=30},CoefficientList[Series[1/2 (x^2 Exp[x])(Cosh[x]^2),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 06 2025 *)
%Y A373065 Cf. A360023.
%K A373065 nonn
%O A373065 0,4
%A A373065 _Enrique Navarrete_, May 21 2024