0,2

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

Comment from Matthijs Coster, Apr 28, 2004: This is the Taylor expansion of a special point on a curve described by Beauville.

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

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

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78.

Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.

Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.

Eric M. Rains, High powers of random elements of compact Lie groups, Probability Theory and Related Fields 107 (1997), 219-241.

T. D. Noe, Table of n, a(n) for n=0..100

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

R. Bacher, Meander algebras

L. Lipshitz and A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 90

(n+1)^2 a_{n+1} = 16n^2 a_{n}. - Matthijs Coster, Apr 28, 2004

a(n) ~ pi^-1*n^-1*2^(4*n) - Joe Keane (jgk(AT)jgk.org), Jun 06 2002

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

E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2x)^2.

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

A002894 := n-> binomial(2*n, n)^2.

CoefficientList[Series[Hypergeometric2F1[1/2, 1/2, 1, 16x], {x, 0, 20}], x]

(PARI) a(n)=binomial(2*n, n)^2

(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 */

(Other) sage: [binomial(2*n, n)**2 for n in xrange(0, 17)] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 21 2009]

Cf. A000984, A000515, A010370, A054474, A060150.

Sequence in context: A163455 A138736 A019999 this_sequence A131765 A132864 A052700

Adjacent sequences: A002891 A002892 A002893 this_sequence A002895 A002896 A002897

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).