A360036 Expansion of e.g.f. x*exp(x)*(sinh(x))^2.

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

Author

Enrique Navarrete, Jan 22 2023

Keywords

• nonn
• easy

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.

Examples

			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)}.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-7,-12,17,6,-9).

Crossrefs

Cf. A081251, A015518, A360023, A360035.

Programs

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 *)

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