A360035 - OEIS

%I #27 Feb 12 2023 15:19:11

%S 0,0,2,6,28,100,366,1274,4376,14760,49210,162382,531444,1727180,

%T 5580134,17936130,57395632,182948560,581130738,1840247318,5811307340,

%U 18305618100,57531942622,180441092746,564859072968,1765184603000,5507375961386,17157594341214,53379182394916

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

%C 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).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,2,-12,-9).

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

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

%F G.f.: 2*x^2*(1 - x)/((1 + x)^2*(1 - 3*x)^2). - _Stefano Spezia_, Jan 23 2023

%e For n = 3, the 6 cases are (where the element selected from the third set is in parenthesis):

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

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

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

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

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

%e {}, {3}, {1, (2)}.

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

%Y Cf. A360023, A360036.

%K nonn,easy

%O 0,3

%A _Enrique Navarrete_, Jan 22 2023