A214624 Braid numbers B((2)^n->(2)^n).

1, 1, 16, 504, 28800, 2620800, 348364800, 63707212800, 15343379251200, 4707627724800000, 1792664637603840000, 829619584788234240000, 458592296933263933440000, 298435681233688170332160000, 225843218230899155927040000000, 196652982274555440023470080000000

Offset: 0

Author

Johan de Ruiter, Jul 23 2012

Keywords

• nonn
• easy

Comments

The number of different possible outcomes when starting with n piles of 2 distinct playing cards and repeatedly moving a top card from either of these n piles to one of n new piles, until all new piles have height 2.

Links

• J. de Ruiter, Counting Classes of Klondike Solitaire Configurations, Master's Thesis (2012), 48-58.

Programs

PARI

a(n) = (2*n)!*(3*n-2)/(4*n-2); \\ Michel Marcus, Aug 18 2013

Formula

a(n) = (2*n)!-n^2*(2*n-2)! for n>0.

a(n) = (2*n)!*(3*n-2)/(4*n-2).

a(n) = a(n-1)*2*n*(2*n-3)*(3*n-2)/(3*n-5) for n>0.

a(n) = Sum_{i=1..n} a(n-i)*C(n,i)*C(n-1,i-1)*i!*(i-1)!*(2^(2*i-1)-1) for n>0.

a(n) = Sum_{i=0..n-1} a(i)*n!*(n-1)!*(2^(2*n-2*i-1)-1)/(i!)^2 for n>0. [corrected by Jason Yuen, Oct 27 2024]

a(n) = Sum_{i=0..n-1} a(i)*((2*n)!!*(2*n-2)!!/((2*i)!!)^2-n!*(n-1)!/(i!)^2) for n>0. [corrected by Jason Yuen, Oct 27 2024]

Extensions

More terms from Michel Marcus, Aug 18 2013