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%I #25 Feb 12 2023 15:19:02
%S 0,0,0,6,24,100,360,1274,4368,14760,49200,162382,531432,1727180,
%T 5580120,17936130,57395616,182948560,581130720,1840247318,5811307320,
%U 18305618100,57531942600,180441092746,564859072944,1765184603000,5507375961360,17157594341214,53379182394888
%N Expansion of e.g.f. x*exp(x)*(sinh(x))^2.
%C 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 odd number of elements and an element is selected from the third.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7,-12,17,6,-9).
%F a(n) = n*A081251(n-2) for n >= 3.
%F a(n) = n*(3^(n-1) + (-1)^(n-1) - 2)/4.
%F G.f.: 2*x^3*(3 - 6*x - x^2)/((1 - x)^2*(1 + x)^2*(1 - 3*x)^2). - _Stefano Spezia_, Jan 23 2023
%e The first 4 cases are shown below for a(4)=24 (where the element selected from the third set is in parenthesis):
%e {1}, {2}, {(3), 4}
%e {1}, {2}, {3, (4)}
%e {2}, {1}, {(3), 4}
%e {2}, {1}, {3, (4)}.
%Y Cf. A081251, A015518, A360023, A360035.
%K nonn,easy
%O 0,4
%A _Enrique Navarrete_, Jan 22 2023
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