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A360023

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

0, 1, 2, 9, 28, 105, 366, 1281, 4376, 14769, 49210, 162393, 531444, 1727193, 5580134, 17936145, 57395632, 182948577, 581130738, 1840247337, 5811307340, 18305618121, 57531942622, 180441092769, 564859072968, 1765184603025, 5507375961386, 17157594341241, 53379182394916, 165856745298489

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OFFSET

0,3

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 an element is selected from the third.

LINKS

Winston de Greef, Table of n, a(n) for n = 0..2080

Index entries for linear recurrences with constant coefficients, signature (6,-7,-12,17,6,-9).

FORMULA

a(n) = n*A122983(n-1) for n > 0.

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

G.f.: x*(1 - 4*x + 4*x^2 + 7*x^4)/((1 - x)^2*(1 + x)^2*(1 - 3*x)^2). - Stefano Spezia, Jan 23 2023

EXAMPLE

For n = 3, the 9 cases are (where the element selected from the third set is in parenthesis):

{}, {}, {(1), 2, 3}

{}, {}, {1, (2), 3}

{}, {}, {1, 2, (3)}