A360035 Expansion of e.g.f. x*exp(x)*cosh(x)*sinh(x).

0, 0, 2, 6, 28, 100, 366, 1274, 4376, 14760, 49210, 162382, 531444, 1727180, 5580134, 17936130, 57395632, 182948560, 581130738, 1840247318, 5811307340, 18305618100, 57531942622, 180441092746, 564859072968, 1765184603000, 5507375961386, 17157594341214, 53379182394916

Offset: 0

Author

Enrique Navarrete, Jan 22 2023

Keywords

• nonn
• easy

Comments

a(n) is the number of ordered set partitions of an n-set into 3 sets such that the first set has an even number of elements, the second set has an odd number of elements, and an element is selected from the third (see example).

Examples

			For n = 3, the 6 cases are (where the element selected from the third set is in parenthesis):
{}, {1}, {(2), 3}
{}, {1}, {2, (3)}
{}, {2}, {(1), 3}
{}, {2}, {1, (3)}
{}, {3}, {(1), 2}
{}, {3}, {1, (2)}.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (4,2,-12,-9).

Crossrefs

A015518 is the case of no element selected in the 3rd set.

Cf. A360023, A360036.

Formula

a(n) = n*A015518(n-1) for n > 0.

a(n) = n*(3^(n-1) - (-1)^(n-1))/4.

G.f.: 2*x^2*(1 - x)/((1 + x)^2*(1 - 3*x)^2). - Stefano Spezia, Jan 23 2023