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)}
{1,2}, {}, {(3)}
{1,3}, {}, {(2)}
{2,3}, {}, {(1)}
{}, {1,2}, {(3)}
{}, {1,3}, {(2)}
{}, {2,3}, {(1)}.
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Jan 22 2023
STATUS
approved