A373065 Expansion of e.g.f. (1/2)*(x^2*exp(x))*(cosh(x))^2.

0, 0, 1, 3, 18, 70, 315, 1281, 5124, 19692, 73845, 270655, 974358, 3454386, 12090351, 41851005, 143489160, 487862872, 1646537193, 5520742011, 18402473370, 61018727070, 201361799331, 661617340153, 2165293113228, 7060738412100, 22947399839325, 74349575478711, 240206320777374, 773998144726282

Offset: 0

Author

Enrique Navarrete, May 21 2024

Keywords

• nonn

Comments

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} .

Examples

			For n = 5, we have the following cases (allowing empty sets):
  {}, {1,2}, {(3), (4), 5}      (30 of these),
  {1,2}, {}, {(3), (4), 5}      (30 of these),
  {}, {}, {(1), (2), 3, 4, 5}   (10 of these),
where the two elements selected from the third set are in parentheses.
		

Crossrefs

Cf. A360023.

Programs

Mathematica

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 *)

Formula

a(n) = binomial(n,2)*(3^(n-2) + (-1)^n + 2)/4.