%I #18 Jun 17 2024 20:03:30
%S 0,0,1,3,18,70,315,1281,5124,19692,73845,270655,974358,3454386,
%T 12090351,41851005,143489160,487862872,1646537193,5520742011,
%U 18402473370,61018727070,201361799331,661617340153,2165293113228,7060738412100,22947399839325,74349575478711,240206320777374,773998144726282
%N Expansion of e.g.f. (1/2)*(x^2*exp(x))*(cosh(x))^2.
%C 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 a(n) = binomial(n,2)*(3^(n-2) + (-1)^n + 2)/4.
%e For n = 5, we have the following cases (allowing empty sets):
%e {}, {1,2}, {(3), (4), 5} (30 of these),
%e {1,2}, {}, {(3), (4), 5} (30 of these),
%e {}, {}, {(1), (2), 3, 4, 5} (10 of these),
%e where the two elements selected from the third set are in parentheses.
%Y Cf. A360023.
%K nonn
%O 0,4
%A _Enrique Navarrete_, May 21 2024