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A360023 Expansion of e.g.f. x*exp(x)*(cosh(x))^2.

%I A360023 #41 Feb 20 2023 14:51:17
%S A360023 0,1,2,9,28,105,366,1281,4376,14769,49210,162393,531444,1727193,
%T A360023 5580134,17936145,57395632,182948577,581130738,1840247337,5811307340,
%U A360023 18305618121,57531942622,180441092769,564859072968,1765184603025,5507375961386,17157594341241,53379182394916,165856745298489
%N A360023 Expansion of e.g.f. x*exp(x)*(cosh(x))^2.
%C A360023 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.
%H A360023 Winston de Greef, <a href="/A360023/b360023.txt">Table of n, a(n) for n = 0..2080</a>
%H A360023 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7,-12,17,6,-9).
%F A360023 a(n) = n*A122983(n-1) for n > 0.
%F A360023 a(n) = n*(3^(n-1) + (-1)^(n-1) + 2)/4.
%F A360023 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
%e A360023 For n = 3, the 9 cases are (where the element selected from the third set is in parenthesis):
%e A360023 {}, {}, {(1), 2, 3}
%e A360023 {}, {}, {1, (2), 3}
%e A360023 {}, {}, {1, 2, (3)}
%e A360023 {1,2}, {}, {(3)}
%e A360023 {1,3}, {}, {(2)}
%e A360023 {2,3}, {}, {(1)}
%e A360023 {}, {1,2}, {(3)}
%e A360023 {}, {1,3}, {(2)}
%e A360023 {}, {2,3}, {(1)}.
%Y A360023 Cf. A122983, A360035, A360036.
%K A360023 nonn,easy
%O A360023 0,3
%A A360023 _Enrique Navarrete_, Jan 22 2023