%I A002894 M3664 N1490
%S A002894 1,4,36,400,4900,63504,853776,11778624,165636900,2363904400,
%T A002894 34134779536,497634306624,7312459672336,108172480360000,
%U A002894 1609341595560000,24061445010950400,361297635242552100
%N A002894 Binomial(2n,n)^2.
%C A002894 a(n) is the number of monotonic paths (only moving N and E) in the lattice 
               [0..2n] X [0..2n] that contain the points (0,0), (n,n) and (2n,2n). 
               - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
%C A002894 Comment from Matthijs Coster, Apr 28, 2004: This is the Taylor expansion 
               of a special point on a curve described by Beauville.
%C A002894 Expansion of K(k)/(pi/2) in powers of m/16=(k/4)^2, where K(k) is complete 
               elliptic integral of first kind evaluated at k. - Michael Somos, 
               Mar 04 2003
%C A002894 Square lattice walks that start and end at origin after 2n steps. - Gareth 
               McCaughan (gareth.mccaughan(AT)pobox.com) and Michael Somos Jun 12 
               2004
%C A002894 If A is a random matrix in USp(4) (4 X 4 X 4omplex matrices that are 
               unitary and symplectic) then a(n)=E[(tr(A^k))^{2n}] for any k > 4. 
               - Andrew V. Sutherland (drew(AT)math.mit.edu), Apr 01 2008
%D A002894 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002894 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002894 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 591,828.
%D A002894 E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes 
               in a rectangle, Discr. Math., 298 (2005). 62-78.
%D A002894 Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre 
               fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, 
               May 24 1982.
%D A002894 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
%D A002894 Matthijs Coster, Over 6 families van krommen [On 6 families of curves], 
               Master's Thesis (unpublished), Aug 26 1983.
%D A002894 C. Domb, On the theory of cooperative phenomena in crystals, Advances 
               in Phys., 9 (1960), 149-361.
%D A002894 Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials 
               and random matrices", preprint, 2008.
%D A002894 Eric M. Rains, High powers of random elements of compact Lie groups, 
               Probability Theory and Related Fields 107 (1997), 219-241.
%H A002894 T. D. Noe, <a href="b002894.txt">Table of n, a(n) for n=0..100</a>
%H A002894 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A002894 R. Bacher, <a href="http://www-fourier.ujf-grenoble.fr/cgi-bin/qau_prep.pl?AutorName=BACHER">
               Meander algebras</a>
%H A002894 L. Lipshitz and A. J. van der Poorten, <a href="http://www-centre.ics.mq.edu.au/
               alfpapers/a084.pdf">Rational functions, diagonals, automata and arithmetic</
               a>
%H A002894 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LatticePath.html">Link to a section of The World of Mathematics.</
               a>
%H A002894 Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/
               abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</
               a>.
%H A002894 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               90
%F A002894 (n+1)^2 a_{n+1} = 16n^2 a_{n}. - Matthijs Coster, Apr 28, 2004
%F A002894 a(n) ~ pi^-1*n^-1*2^(4*n) - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
%F A002894 G.f.: F(1/2, 1/2;1;16x) = 1/AGM(1, (1-16x)^(1/2)) = K(4sqrt(x))/(pi/2), 
               where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre. 
               - Michael Somos, Mar 04 2003
%F A002894 E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2x)^2.
%F A002894 a(n) = A000984(n)^2 = ((2*n)!/(n!)^2)^2 = (((2*n)!)^2)/((n!)^4). a(n) 
               = A000984(n)^2 = ((((2^n)*(2*n-1)!!)/(n!)))^2 = (((2^(2*n))*(2*n-1)!!)^2)/
               (n!)^2). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 17 2007
%p A002894 A002894 := n-> binomial(2*n,n)^2.
%t A002894 CoefficientList[Series[Hypergeometric2F1[1/2, 1/2, 1, 16x], {x, 0, 20}], 
               x]
%o A002894 (PARI) a(n)=binomial(2*n,n)^2
%o A002894 (PARI) a(n)=if(n<0,0,polcoeff(polcoeff(polcoeff(1/(1-x*(y+z+1/y+1/z))+x*O(x^(2*n)),
               2*n),0),0)) /* Michael Somos Jun 12 2004 */
%o A002894 (Other) sage: [binomial(2*n,n)**2 for n in xrange(0, 17)] [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 21 2009]
%Y A002894 Cf. A000984, A000515, A010370, A054474, A060150.
%Y A002894 Sequence in context: A163455 A138736 A019999 this_sequence A131765 A132864 
               A052700
%Y A002894 Adjacent sequences: A002891 A002892 A002893 this_sequence A002895 A002896 
               A002897
%K A002894 nonn,easy,nice
%O A002894 0,2
%A A002894 N. J. A. Sloane (njas(AT)research.att.com).