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

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

0,5

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-24,0,78,-54,-80,72,27,-27).

Formula

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

G.f.: 4*x^4*(3 - 12*x + 12*x^2 + 5*x^4)/((1 - x)^3*(1 + x)^3*(1 - 3*x)^3). - Andrew Howroyd, Nov 13 2025

Example

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.

Mathematica

A373134[n_] := n*(n-1)*(3^(n-2) + (-1)^n - 2)/8; Array[A373134, 30, 0] (* Paolo Xausa, Jan 19 2026 *)

Crossrefs

Cf. A360036, A373065

Sequence in context: A374648 A074433 A280175 * A216491 A114307 A009031

Adjacent sequences: A373131 A373132 A373133 * A373135 A373136 A373137

Keyword

nonn

Author

Enrique Navarrete, May 26 2024

Status

approved