0,2

a(n) = number of walks of 2n unit steps North, East, South, or West, starting at the origin, bounded above by y=x, below by y=-x and terminating on the ray y=x>=0. Example: a(1) counts EN, EW; a(2) counts ESNN, ESNW, ENSN, ENSW, ENEN, ENEW, EENN, EENW, EEWN, EEWW, EWEN, EWEW. - David Callan (callan(AT)stat.wisc.edu), Oct 11 2005

Bijective proof: given such a NESW walk, construct a pair (P_1, P_2) of lattice paths of upsteps U=(1,1) and downsteps D=(1,-1) as follows. To get P_1, replace each E and S by U and each W and N by D. To get P_2, replace each N and E by U and each S and W by D. For example, EENSNW -> (UUDUDD, UUUDUD). This mapping is 1-to-1 and its range is the Cartesian product of the set of Dyck n-paths and the set of nonnegative paths of length 2n. The Dyck paths are counted by the Catalan number C_n (A000108) and the nonnegative paths are counted (see for example the Callan link) by the central binomial coefficient binom(2n,n) (A000984). So this is a bijection from these NESW walks to a set of size C_n*binom(2n,n)=a(n). - David Callan (callan(AT)stat.wisc.edu), Sep 18 2007

If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic), then a(n)=E[(tr(A^3))^{2n}]. - Andrew V. Sutherland (drew(AT)math.mit.edu), Apr 01 2008

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 93.

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

T. M. Macrobert, Functions of a Complex Variable, 4th ed., Macmillan & Co, London, 1958, p. 177

David Callan, Bijections for the identity 4^n = ...

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

G.f.: 1/4*((16*x-1)*EllipticK(4*x^(1/2))+EllipticE(4*x^(1/2)))/x/Pi. - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 12 2003

Given G.f. A(x), y=xA(x) satisfies y=y''*(1-16x)x/4. - Michael Somos Sep 11 2005

Binomial(2*n,n)^2/(n+1) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 27 2006

G.f.: F(1/2,1/2;2;16x); [From Paul Barry (pbarry(AT)wit.ie), Sep 03 2008]

[seq(binomial(2*n, n)^2/(n+1), n=0..17)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 27 2006

(PARI) a(n)=if(n<0, 0, (2*n)!^2/n!^4/(n+1)) /* Michael Somos Sep 11 2005 */

A002894(n)=(n+1)a(n).

Cf. A000108.

Sequence in context: A009816 A064370 A138421 this_sequence A151392 A079821 A124102

Adjacent sequences: A000885 A000886 A000887 this_sequence A000889 A000890 A000891

nonn

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