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

0,3

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).

Links

Table of n, a(n) for n=0..28.

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

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

Example

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)}.

Crossrefs

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

Cf. A360023, A360036.

Sequence in context: A322507 A340471 A242511 * A323268 A089748 A047125

Adjacent sequences: A360032 A360033 A360034 * A360036 A360037 A360038

Keyword

nonn,easy

Author

Enrique Navarrete, Jan 22 2023

Status

approved