A360035 -id:A360035 - cp's OEIS frontend

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

0, 1, 2, 9, 28, 105, 366, 1281, 4376, 14769, 49210, 162393, 531444, 1727193, 5580134, 17936145, 57395632, 182948577, 581130738, 1840247337, 5811307340, 18305618121, 57531942622, 180441092769, 564859072968, 1765184603025, 5507375961386, 17157594341241, 53379182394916, 165856745298489

Offset: 0

Enrique Navarrete, Jan 22 2023

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.

			For n = 3, the 9 cases are (where the element selected from the third set is in parenthesis):
{}, {}, {(1), 2, 3}
{}, {}, {1, (2), 3}
{}, {}, {1, 2, (3)}
{1,2}, {}, {(3)}
{1,3}, {}, {(2)}
{2,3}, {}, {(1)}
{}, {1,2}, {(3)}
{}, {1,3}, {(2)}
{}, {2,3}, {(1)}.

Winston de Greef, Table of n, a(n) for n = 0..2080
Index entries for linear recurrences with constant coefficients, signature (6,-7,-12,17,6,-9).

Cf. A122983, A360035, A360036.

a(n) = n*A122983(n-1) for n > 0.
a(n) = n*(3^(n-1) + (-1)^(n-1) + 2)/4.
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

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

Original entry on oeis.org

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

Enrique Navarrete, Jan 22 2023

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.

			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)}.

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

Cf. A081251, A015518, A360023, A360035.

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 *)

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

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

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

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

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

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

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