%I #68 Jul 04 2023 09:32:51
%S 1,1,4,9,36,100,400,1225,4900,15876,63504,213444,853776,2944656,
%T 11778624,41409225,165636900,590976100,2363904400,8533694884,
%U 34134779536,124408576656,497634306624,1828114918084
%N a(n) = binomial(n, floor(n/2))^2 = A001405(n)^2.
%C a(n) is also the number of rooted two-vertex (or, dually, two-face) regular planar maps of valency n+1. - _Valery A. Liskovets_, Oct 19 2005
%C If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic), then a(n)=(-1)^n*E[(tr(A^4))^n]. - _Andrew V. Sutherland_, Apr 01 2008
%H Stephanie Anderson, <a href="/A018224/b018224.txt">Table for n, a(n) for n = 0..200</a>
%H Alin Bostan, <a href="https://www-apr.lip6.fr/sem-comb-slides/IHP-bostan.pdf">Computer Algebra for Lattice Path Combinatorics</a>, Slides, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
%H Alin Bostan, <a href="https://semflajolet.math.cnrs.fr/wp-content/uploads/2022/03/Bostan-synthese-IHP.pdf">Computer Algebra for Lattice Path Combinatorics</a>, Synthesis, Séminaire de Combinatoire Ph. Flajolet, June 06 2013.
%H Alin Bostan, <a href="https://specfun.inria.fr/bostan/HDR.pdf">Calcul Formel pour la Combinatoire des Marches</a> [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017.
%H Alin Bostan, Frédéric Chyzak, Mark van Hoeij, Manuel Kauers, and Lucien Pech, <a href="https://doi.org/10.1016/j.ejc.2016.10.010">Hypergeometric expressions for generating functions of walks with small steps in the quarter plane</a>, Eur. J. Comb. 61, 242-275 (2017)
%H M. Bousquet, G. Labelle and P. Leroux, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00406-9">Enumeration of planar two-face maps</a>, Discrete Math., vol. 222 (2000), 1-25.
%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</a>, arXiv:0803.4462 [math.NT], 2008-2010.
%H Helmut Prodinger, <a href="https://arxiv.org/abs/1911.07604">Two New Identities Involving the Catalan Numbers: A classical approach</a>, arXiv:1911.07604 [math.CO], 2019.
%F E.g.f.: BesselI(0, 2*x)*(BesselI(0, 2*x)+BesselI(1, 2*x)). - _Vladeta Jovovic_, Jun 12 2005
%F G.f. (1+1/(4*x))*hypergeom([1/2, 1/2],[1],16*x^2)-1/(4*x). - _Mark van Hoeij_, Oct 13 2009
%F a(n) = (n!/(floor(n/2)!*floor((n+1)/2)!))^2. - _Peter Luschny_, Apr 29 2014
%F a(n) = A056040(n) * A056040(n+1) / (n+1). - _Peter Luschny_, Apr 29 2014
%F a(n) = 4^n*[x^n]((1+x)*EllipticK(x)/(x*Pi/2)-1/x). - _Peter Luschny_, Oct 14 2015
%F a(n) ~ 4^n*((2*n+3)/(2*n+1))^((-1)^n/2)/((n+1)*Pi/2). - _Peter Luschny_, Oct 14 2015
%F a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*C(k)*binomial(2*n-2*k,n-k) where C(k) are Catalan numbers (A000108), see Prodinger. - _Michel Marcus_, Nov 19 2019
%F From _Peter Bala_, Jul 03 2023: (Start)
%F Right hand side of the binomial sum identity (1/2)*Sum_{k = 0..n+1} (-1)^k*4^(n+1-k)*binomial(n+1,k)*binomial(n+k,k)*binomial(2*k,k) = a(n).
%F a(n) = (1/2)*4^(n+1) * hypergeom([n+1, -n-1, 1/2], [1, 1], 1).
%F P-recursive:
%F (2*n - 1)*(n + 1)^2*a(n) = 4*(2*n^2 - 1)*a(n-1) + 16*(2*n + 1)*(n - 1)^2*a(n-2) with a(0) = a(1) = 1. (End)
%p s := x -> (1+x)*EllipticK(x)/(x*Pi/2)-1/x:
%p seq(4^n*coeff(series(s(x),x,n+2),x,n),n=0..23); # _Peter Luschny_, Oct 14 2015
%t (* Note that Mathematica uses a different definition of the EllipticK function. *)
%t CoefficientList[Series[(-Pi + (2 + 8 x) EllipticK[16 x^2])/(4 Pi x), {x,0,23}], x] (* _Peter Luschny_, Oct 14 2015 *)
%t Table[Binomial[n,Floor[n/2]]^2,{n,0,30}] (* _Harvey P. Dale_, Dec 02 2022 *)
%o (PARI) vector(50, n, n--; binomial(n, n\2)^2) \\ _Altug Alkan_, Oct 14 2015
%Y Bisections are A002894 and A060150.
%Y Cf. A000108, A001405, A056040, A113182, A138545.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_