A364422 Expansion of e.g.f. x^2/(1-x-x^2)^2.

0, 0, 2, 12, 120, 1200, 14400, 191520, 2862720, 47174400, 852768000, 16765056000, 356377190400, 8144943206400, 199202395392000, 5191467240960000, 143634952581120000, 4204936774950912000, 129865748246986752000, 4219868533182382080000, 143918318293689139200000

Offset

0,3

Comments

a(n) is the number of ways to partition [n] into blocks of size at most 2, order the blocks, order the elements within each block, and choose 2 elements from a block.

Links

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

Formula

a(n) = A000142(n)*A001629(n).

Example

a(6)=14400 since the number of ways to partition [6] into blocks of size at most 2, order the blocks, order the elements within each block, and select 2 elements from a block are the following:
  (i) 12,3,4,5,6: 3600 such orderings, 1 way to choose 2 elements (from the block with 2 elements), hence 3600 ways;
  (ii) 12,34,5,6: 4320 such orderings, 2 ways to choose 2 elements (from one of the two blocks with 2 elements), hence 8640 ways.
  (iii) 12,34,56: 720 such orderings, 3 ways to choose 2 elements (from one of the three blocks with 2 elements), hence 2160 ways.

Mathematica

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

Crossrefs

Cf. A000045, A000142, A001629, A005442.

Sequence in context: A089431 A119701 A362457 * A286629 A361595 A370876

Adjacent sequences: A364419 A364420 A364421 * A364423 A364424 A364425

Keyword

nonn

Author

Enrique Navarrete, Sep 01 2023

Status

approved