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%I #82 Mar 31 2023 15:18:51
%S 2,4,12,40,140,504,1848,6864,25740,97240,369512,1410864,5408312,
%T 20801200,80233200,310235040,1202160780,4667212440,18150270600,
%U 70690527600,275693057640,1076515748880,4208197927440,16466861455200
%N Twice central binomial coefficients.
%C Central elements in the even-Pascal triangle A028326.
%C If Y is a 3-subset of an 2n-set X then, for n>=3, a(n-1) is the number of (n+1)-subsets of X having at least two elements in common with Y. - _Milan Janjic_, Dec 16 2007
%C a(n) denotes the number of ways one can reach the (n,n) point in an n X n grid via the point (n-1, n-1) starting from (0,0) when moving right and up is allowed [From Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 29 2009]
%C It appears that a(n-1) is also the number of quivers in the mutation class of twisted types BD_n and CD_n for n >= 3. - _Christian Stump_, Nov 03 2010
%C This is the case m = n+1 in the Catalan's formula (2m)!*(2n)!/(m!*(m+n)!*n!) - see Umberto Scarpis in References. - _Bruno Berselli_, Apr 27 2012
%C a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that bounce off the diagonal y = x an even number of times. Details can be found in Section 4.2 in Pan and Remmel's link. - _Ran Pan_, Feb 01 2016
%C a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that cross the diagonal y = x an even number of times. Details can be found in Section 4.3 in Pan and Remmel's link. - _Ran Pan_, Feb 01 2016
%D Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third Edition), page 11.
%H Vincenzo Librandi, <a href="/A028329/b028329.txt">Table of n, a(n) for n = 0..300</a>
%H Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, <a href="https://arxiv.org/abs/2301.09765">Enumeration of multi-rooted plane trees</a>, arXiv:2301.09765 [math.CO], 2023.
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H Ran Pan and Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%F G.f.: 2/sqrt(1 - 4*x).
%F a(n) = 2*A000984(n).
%F a(n) = 2 * binomial(2*n, n).
%F a(n) = A100320(n) = A095660(2*n,n) for n > 0. - _Reinhard Zumkeller_, Apr 08 2012
%F G.f.: G(0), where G(k)= 1 + 1/(1 - 2*x*(2*k + 1)/(2*x*(2*k + 1) + (k + 1)/ G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 07 2013
%F a(n) = binomial(2*n+2, n+1) - A162551(n). - _Ran Pan_, Feb 01 2016
%F D-finite with recurrence: n*a(n) + 2*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020
%p seq(add(binomial(2*n,n),k=1..2),n=0..23); # _Zerinvary Lajos_, Dec 14 2007
%t Table[2Binomial[2n,n],{n,0,30}] (* _Harvey P. Dale_, Aug 08 2011 *)
%o (PARI) a(n)=2*binomial(2*n,n)
%Y Bisection of A047073, A063886.
%Y First differences of A054113.
%Y Cf. A000984, A095660, A100320, A162551.
%K nonn,easy
%O 0,1
%A _Mohammad K. Azarian_
%E Edited by _Michael Somos_, Sep 13 2003
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