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%I #26 Feb 09 2022 09:05:44
%S 1,2,16,128,1156,10952,107584,1083392,11115556,115702472,1218289216,
%T 12948910592,138708574096,1495661223968,16218468710656,
%U 176727219273728,1933956651447076,21243204576601928,234121111199439424,2587943032046002688
%N Braid numbers B((n,n)->(n,n)).
%C 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.
%C For even values of n, a(n) is a square, while for odd values of n, a(n) is twice a square.
%H J. de Ruiter, <a href="http://www.liacs.nl/assets/Masterscripties/2012-09JohandeRuiter.pdf">Counting Classes of Klondike Solitaire Configurations</a>, Master's Thesis (2012), 48-58.
%F 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
%F a(n) = (-2)^n*hypergeom([1/2, -n, n + 1], [1, 1], 2). - _Peter Luschny_, Mar 14 2018
%F a(n) ~ 2^(n - 3/2) * (1 + sqrt(2))^(2*n + 1) / (Pi*n). - _Vaclav Kotesovec_, Jun 09 2019
%F G.f.: Sum_{n >= 0} binomial(2*n,n)^2*x^n/(1 + 2*x)^(2*n+1). - _Peter Bala_, Feb 07 2022
%t a[n_] := (-2)^n HypergeometricPFQ[{1/2, -n, n + 1}, {1, 1}, 2];
%t Table[a[n], {n, 0, 19}] (* _Peter Luschny_, Mar 14 2018 *)
%K nonn
%O 0,2
%A _Johan de Ruiter_, Jul 23 2012
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