A386961 Expansion of e.g.f. (x^2/(1-x)^3)*exp(x/(1-x)).

0, 0, 2, 24, 252, 2720, 31350, 389592, 5224184, 75411072, 1167800490, 19329325400, 340729559412, 6374829580704, 126189319150622, 2635193467383960, 57900808243004400, 1335311847609293312, 32250677891048663634, 814075526093792723352, 21435972525439506572780, 587788552746385612893600

Offset

0,3

Comments

Number of ways to sit n people on benches and choose 2 from any bench (different from choosing 2 from each bench, which is A386514).

If only 1 bench is used the sequence is A001804.

Links

Table of n, a(n) for n=0..21.

Formula

From Vaclav Kotesovec, Sep 13 2025: (Start)

Recurrence: (n-2)*a(n) = 2*(n-1)*n*a(n-1) - (n-1)^2*n*a(n-2).

a(n) ~ n^(n + 5/4) / (sqrt(2) * exp(n - 2*sqrt(n) + 1/2)) * (1 - 65/(48*sqrt(n))). (End)

a(n) = (n-1) * n! * LaguerreL(n-2, -1). - Mark van Hoeij, Jan 14 2026

Example

a(5)=2720 since the number of ways are (number of people in parentheses):
  1 bench (5): 120 configurations, 1200 ways;
  2 benches (4,1): 120 configurations, 720 ways;
  2 benches (3,2): 120 configurations, 480 ways;
  3 benches (3,1,1): 60 configurations, 180 ways;
  3 benches (2,2,1): 60 configurations, 120 ways;
  4 benches (2,1,1,1): 20 configurations, 20 ways;
  5 benches (1,1,1,1,1): 1 configuration, 0 ways, for a total of 2720 ways.

Mathematica

CoefficientList[Series[(x^2/(1-x)^3)*Exp[x/(1-x)], {x, 0, 21}], x]*Range[0, 21]! (* Stefano Spezia, Sep 11 2025 *)

Crossrefs

Cf. A000262, A001804, A386514.

Sequence in context: A187584 A386702 A173228 * A387244 A061190 A300399

Adjacent sequences: A386958 A386959 A386960 * A386962 A386963 A386964

Keyword

nonn

Author

Enrique Navarrete, Sep 10 2025

Status

approved