A387083 Expansion of e.g.f. (2*(1-x)^2)/(2-4*x+x^2).

1, 0, 1, 6, 42, 360, 3690, 44100, 602280, 9253440, 157966200, 2966317200, 60765843600, 1348539192000, 32229405608400, 825285553092000, 22541609025936000, 654175871661312000, 20101465198839024000, 651991603501798560000, 22260385752292527840000

Offset: 0

Author

Enrique Navarrete, Aug 16 2025

Keywords

• nonn
• new

Comments

For n > 0, a(n) is the number of ways to sit n people around circular tables, choose 2 people from each table, and linear order the tables.

Examples

			a(6)=3690 since for 6 people the number of ways to do the combined tasks in the comment are: 1800 using one table, 1080 using two tables with 4 and 2 people, 720 using two tables with 3 people each, and 90 using three tables with 2 people each.
		

Crossrefs

Cf. A001286.

Programs

Mathematica

With[{m = 20}, CoefficientList[Series[(2*(1 - x)^2)/(2 - 4*x + x^2), {x, 0, m}], x] * Table[n!, {n, 0, m}]] (* Amiram Eldar, Aug 16 2025 *)

Formula

For n > 0, a(n) = ((1 + sqrt(2))^(n-1) - (sqrt(2) - 1)^(n-1)) * n! / 2^(n/2 + 1). - Vaclav Kotesovec, Aug 18 2025

D-finite with recurrence 2*a(n) -4*n*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Aug 26 2025

a(n) = A060995(n)*n!/2^n. - R. J. Mathar, Aug 26 2025