%I #68 Nov 02 2024 18:00:35
%S 1,1,9,100,1225,15876,213444,2944656,41409225,590976100,8533694884,
%T 124408576656,1828114918084,27043120090000,402335398890000,
%U 6015361252737600,90324408810638025,1361429497505672100,20589520178326522500,312321918272897610000
%N a(0) = 1; for n > 0, binomial(2n-1, n-1)^2.
%C Number of square lattice walks that start at (0,0) and end at (1,0) after 2n-1 steps, free to pass through (1,0) at intermediate steps. - _Steven Finch_, Dec 20 2001
%C Number of paths of length n connecting two neighboring nodes in optimal chordal graph of degree 4, G(2*d(G)^2+2*d(G)+1,2d(G)+1), of diameter d(G). - B. Dubalski (dubalski(AT)atr.bydgoszcz.pl), Feb 05 2002
%C a(n) is the number of ways to place n red balls and n blue balls into n distinguishable boxes with no restrictions on the number of balls put in a box. - _Geoffrey Critzer_, Jul 08 2013
%C The number of square lattice walks of n steps that start at the origin and end at (k,0) is zero if n-k is odd and [binomial(n,(n-k)/2)]^2 if n-k is even. - _R. J. Mathar_, Sep 28 2020
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 1994 Addison-Wesley company, Inc.
%D A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (5.1.29.2)
%D K. A. Ross and C. R. B. Wright, Discrete Mathematics, 1992 Prentice Hall Inc.
%H Harry J. Smith, <a href="/A060150/b060150.txt">Table of n, a(n) for n = 0..200</a>
%H R. Bacher, <a href="https://www-fourier.ujf-grenoble.fr/sites/default/files/ref_478.pdf">Meander algebras</a>
%F a(n) = A088218(n)^2.
%F a(n) = A002894(n)/4 for n>0.
%F G.f.: 1 + (1/AGM(1, sqrt(1-16*x))-1)/4. - _Michael Somos_, Dec 12 2002
%F G.f. = 1 + (K(16x)-1)/4 = 1 + Sum_{k>0} q^k/(1+q^(2k)) where K(16x) is the complete Elliptic integral of the first kind at 16x=k^2 and q is the nome. - _Michael Somos_, May 09 2005
%F G.f.: 1 + x*3F2((1, 3/2, 3/2); (2, 2))(16*x). - _Olivier GĂ©rard_, Feb 16 2011
%F E.g.f.: Sum_{n>0} a(n)*x^(2n-1)/(2n-1)! = BesselI(0, 2x)*BesselI(1, 2x) . - _Michael Somos_, Jun 22 2005
%F D-finite with recurrence n^2*a(n) -4*(2*n-1)^2*a(n-1)=0. - _R. J. Mathar_, Jul 26 2014
%F From _Seiichi Manyama_, Oct 19 2016: (Start)
%F Let the number of multisets of length k on n symbols be denoted by ((n, k)) = binomial(n+k-1, k).
%F a(n) = (Sum_{0 <= k <= n} binomial(n, k)^2 * ((2*n, n - k)))/3 for n > 0. (End)
%F a(n) ~ 4^(2*n-1)/(Pi*n). - _Ilya Gutkovskiy_, Oct 19 2016
%F For n >= 1, a(n) = 1/n * Sum_{k = 0..n-1} (n + 2*k)*binomial(n+k-1, k)^2 = ( 1/(4*n) * Sum_{k = 0..n} (n + 2*k)*binomial(-n+k-1, k)^2 )^2. - _Peter Bala_, Nov 02 2024
%p seq(coeff(series(EllipticK(4*sqrt(x))/(2*Pi) + 3/4, x=0, n+1), x, n), n=0..30); # _Mark van Hoeij_, Apr 30 2013
%t Table[Binomial[2n-1,n]^2,{n,0,19}] (* _Geoffrey Critzer_, Jul 08 2013 *)
%o (PARI) a(n)=if(n<2, 1, binomial(2*n-1,n-1)^2)
%o (PARI) for (n=0, 200, if (n==0, a=1, a=binomial(2*n - 1, n - 1)^2); write("b060150.txt", n, " ", a)) \\ _Harry J. Smith_, Jul 02 2009
%Y Cf. A002894, A135389, A337900, A337901, A337902.
%K nonn,easy,changed
%O 0,3
%A _N. J. A. Sloane_, Apr 10 2001