A360023 -id:A360023 - cp's OEIS frontend

A360036 Expansion of e.g.f. xexp(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

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

A360035 Expansion of e.g.f. xexp(x)cosh(x)*sinh(x).

Original entry on oeis.org

0, 0, 2, 6, 28, 100, 366, 1274, 4376, 14760, 49210, 162382, 531444, 1727180, 5580134, 17936130, 57395632, 182948560, 581130738, 1840247318, 5811307340, 18305618100, 57531942622, 180441092746, 564859072968, 1765184603000, 5507375961386, 17157594341214, 53379182394916

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 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).

			For n = 3, the 6 cases are (where the element selected from the third set is in parenthesis):
{}, {1}, {(2), 3}
{}, {1}, {2, (3)}
{}, {2}, {(1), 3}
{}, {2}, {1, (3)}
{}, {3}, {(1), 2}
{}, {3}, {1, (2)}.

Index entries for linear recurrences with constant coefficients, signature (4,2,-12,-9).

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

Cf. A360023, A360036.

a(n) = n*A015518(n-1) for n > 0.
a(n) = n*(3^(n-1) - (-1)^(n-1))/4.
G.f.: 2*x^2*(1 - x)/((1 + x)^2*(1 - 3*x)^2). - Stefano Spezia, Jan 23 2023

A373065 Expansion of e.g.f. (1/2)(x^2exp(x))*(cosh(x))^2.

Original entry on oeis.org

0, 0, 1, 3, 18, 70, 315, 1281, 5124, 19692, 73845, 270655, 974358, 3454386, 12090351, 41851005, 143489160, 487862872, 1646537193, 5520742011, 18402473370, 61018727070, 201361799331, 661617340153, 2165293113228, 7060738412100, 22947399839325, 74349575478711, 240206320777374, 773998144726282

Offset: 0

Enrique Navarrete, May 21 2024

nonn

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 two elements are selected from the third. "Ordered set partitions", because {}, {1,2}, {(3), (4), 5} is considered to be different from {1,2}, {}, {(3), (4), 5} .

			For n = 5, we have the following cases (allowing empty sets):
  {}, {1,2}, {(3), (4), 5}      (30 of these),
  {1,2}, {}, {(3), (4), 5}      (30 of these),
  {}, {}, {(1), (2), 3, 4, 5}   (10 of these),
where the two elements selected from the third set are in parentheses.

Cf. A360023.

With[{nn=30},CoefficientList[Series[1/2 (x^2 Exp[x])(Cosh[x]^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 06 2025 *)

a(n) = binomial(n,2)*(3^(n-2) + (-1)^n + 2)/4.

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

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A360035 Expansion of e.g.f. xexp(x)cosh(x)*sinh(x).

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A373065 Expansion of e.g.f. (1/2)(x^2exp(x))*(cosh(x))^2.

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

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A360035 Expansion of e.g.f. x*exp(x)*cosh(x)*sinh(x).

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A373065 Expansion of e.g.f. (1/2)*(x^2*exp(x))*(cosh(x))^2.

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

A360035 Expansion of e.g.f. xexp(x)cosh(x)*sinh(x).

A373065 Expansion of e.g.f. (1/2)(x^2exp(x))*(cosh(x))^2.