A214623 Braid numbers B((n,n)->(n,n)).

1, 2, 16, 128, 1156, 10952, 107584, 1083392, 11115556, 115702472, 1218289216, 12948910592, 138708574096, 1495661223968, 16218468710656, 176727219273728, 1933956651447076, 21243204576601928, 234121111199439424, 2587943032046002688

Offset

0,2

Comments

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

For even values of n, a(n) is a square, while for odd values of n, a(n) is twice a square.

Links

Table of n, a(n) for n=0..19.

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

Formula

G.f.: hypergeom([1/12, 5/12],[1],1728/(16*x^4-32*x^3-24*x^2-8*x+1)^3*x^4*(4*x^2-12*x+1)*(2*x+1)^2)/(16*x^4-32*x^3-24*x^2-8*x+1)^(1/4); (based on guessed recurrence). - Mark van Hoeij, Apr 11 2014

a(n) = (-2)^n*hypergeom([1/2, -n, n + 1], [1, 1], 2). - Peter Luschny, Mar 14 2018

a(n) ~ 2^(n - 3/2) * (1 + sqrt(2))^(2*n + 1) / (Pi*n). - Vaclav Kotesovec, Jun 09 2019

G.f.: Sum_{n >= 0} binomial(2*n,n)^2*x^n/(1 + 2*x)^(2*n+1). - Peter Bala, Feb 07 2022

Mathematica

a[n_] := (-2)^n HypergeometricPFQ[{1/2, -n, n + 1}, {1, 1}, 2];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 14 2018 *)

Crossrefs

Sequence in context: A013730 A270924 A371814 * A333719 A161737 A171451

Adjacent sequences: A214620 A214621 A214622 * A214624 A214625 A214626

Keyword

nonn

Author

Johan de Ruiter, Jul 23 2012

Status

approved