A360588 Expansion of e.g.f. (exp(x)-1)^2*(x+x^2/2).

0, 0, 0, 6, 36, 130, 390, 1064, 2744, 6822, 16530, 39292, 92004, 212810, 487214, 1105680, 2490096, 5570254, 12385962, 27393668, 60292700, 132120114, 288357894, 627047896, 1358953896, 2936012150, 6325009730, 13589544204, 29125246164, 62277024922, 132875549790, 282930969632, 601295420384

Offset

0,4

Comments

a(n) is the number of ordered set partitions of an n-set into 3 sets such that the first and second sets contain at least one element and the third set contains either one or two elements.

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Formula

a(n) = n*2^(n-1) + binomial(n,2)*(2^(n-2)-2) - 2*n, n >= 3, a(n) = 0 otherwise.

Example

a(5)=130 since the set partitions are the following:
  20 of the form {1,2,3}, {4}, {5};
  20 of the form {1}, {2,3,4}, {5};
  30 of the form {1,2}, {3,4}, {5};
  30 of the form {1,2}, {3}, {4,5};
  30 of the form {1}, {2,3}, {4,5}.

Mathematica

A360588[n_] := If[n <= 2, 0, 2^(n - 3)*n*(n + 3) - n*(n + 1)];
Array[A360588, 35, 0] (* Paolo Xausa, Jun 25 2026 *)

Prog

(PARI) a(n) = if (n>=3, n*2^(n-1) + binomial(n, 2)*(2^(n-2)-2) - 2*n, 0); \\ Michel Marcus, Feb 13 2023

Crossrefs

Sequence in context: A253945 A331576 A056375 * A321579 A387869 A364429

Adjacent sequences: A360585 A360586 A360587 * A360589 A360590 A360591

Keyword

nonn,easy

Author

Enrique Navarrete, Feb 12 2023

Status

approved