A365330 Expansion of e.g.f. x^3/(1-x-x^2-x^3)^2.

0, 0, 0, 6, 48, 600, 8640, 131040, 2257920, 42819840, 885427200, 19918483200, 483791616000, 12622171161600, 352200296448000, 10466625641472000, 330077933273088000, 11010660024139776000, 387369218691366912000, 14335266857678807040000, 556691771706962411520000

Offset

0,4

Comments

a(n) is the number of ways to partition [n] into blocks of size at most 3, order the blocks, order the elements within each block, and choose 3 elements from a block.

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = A000142(n)*A073778(n+1).

Example

a(6)=8640 since the ways to partition [6] into blocks of size at most 3, order the blocks, order the elements within each block, and select 3 elements from a block are the following:
  (i) 123,4,5,6: 2880 such orderings, 1 way to choose three elements (from the block with 3 elements), hence 2880 ways;
  (ii) 123,45,6: 4320 such orderings, 1 way to choose three elements (from the block with 3 elements), hence 4320 ways;
  (iii) 123,456: 720 such orderings, 2 ways to choose three elements (from one of the two blocks with 3 elements), hence 1440 ways.

Mathematica

With[{m = 20}, Range[0, m]! * CoefficientList[Series[x^3/(1 - x - x^2 - x^3)^2, {x, 0, m}], x]] (* Amiram Eldar, Sep 02 2023 *)

Crossrefs

Cf. A000073, A000142, A189886, A364324, A364422.

Sequence in context: A223270 A376323 A366339 * A052731 A052711 A113388

Adjacent sequences: A365327 A365328 A365329 * A365331 A365332 A365333

Keyword

nonn

Author

Enrique Navarrete, Sep 01 2023

Status

approved