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URL: https://oeis.org/A360036

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A360036
Expansion of e.g.f. x*exp(x)*(sinh(x))^2.
3
0, 0, 0, 6, 24, 100, 360, 1274, 4368, 14760, 49200, 162382, 531432, 1727180, 5580120, 17936130, 57395616, 182948560, 581130720, 1840247318, 5811307320, 18305618100, 57531942600, 180441092746, 564859072944, 1765184603000, 5507375961360, 17157594341214, 53379182394888
OFFSET
0,4
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 odd number of elements and an element is selected from the third.
FORMULA
a(n) = n*A081251(n-2) for n >= 3.
a(n) = n*(3^(n-1) + (-1)^(n-1) - 2)/4.
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
EXAMPLE
The first 4 cases are shown below for a(4)=24 (where the element selected from the third set is in parenthesis):
{1}, {2}, {(3), 4}
{1}, {2}, {3, (4)}
{2}, {1}, {(3), 4}
{2}, {1}, {3, (4)}.
MATHEMATICA
With[{nn=30}, CoefficientList[Series[x Exp[x]Sinh[x]^2, {x, 0, nn}], x] Range[0, nn]!] (* or *) LinearRecurrence[{6, -7, -12, 17, 6, -9}, {0, 0, 0, 6, 24, 100}, 30] (* Harvey P. Dale, Aug 17 2025 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Jan 22 2023
STATUS
approved