cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

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Author

Enrique Navarrete, Jan 22 2023

Keywords

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 even number of elements, and an element is selected from the third.

Examples

			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)}.
		

Crossrefs

Formula

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

A360035 Expansion of e.g.f. x*exp(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

Views

Author

Enrique Navarrete, Jan 22 2023

Keywords

Comments

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).

Examples

			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)}.
		

Crossrefs

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

Formula

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

A373134 Expansion of e.g.f. (1/2)*(x^2*exp(x))*(sinh(x))^2.

Original entry on oeis.org

0, 0, 0, 0, 12, 60, 300, 1260, 5096, 19656, 73800, 270600, 974292, 3454308, 12090260, 41850900, 143489040, 487862736, 1646537040, 5520741840, 18402473180, 61018726860, 201361799100, 661617339900, 2165293112952, 7060738411800, 22947399839000, 74349575478360, 240206320776996, 773998144725876
Offset: 0

Views

Author

Enrique Navarrete, May 26 2024

Keywords

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 two elements are selected from the third.

Examples

			For n = 6, we have the following cases:
   {1}, {2}, {(3), (4), 5, 6} (90 of these),
   {2}, {1}, {(3), (4), 5, 6} (90 of these),
   {1, 2, 3}, {4}, {(5), (6)} (60 of these),
   {1}, {2, 3, 4}, {(5), (6)} (60 of these),
where the two elements selected from the third set are in parentheses.
		

Crossrefs

Formula

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