0,2

Number of walks with 2n steps on the cubic lattice Z x Z x Z beginning and ending at (0,0,0)).

If A is a random matrix in USp(6) (6 X 6 complex matrices that are unitary and symplectic) then a(n) is the 2nth moment of tr(A^k) for all k >= 7. - Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 24 2008

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891.

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

J. A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, Journal of Statistical Computation and Simulation, 51 (1995), 291-313.

G. S. Joyce, The simple cubic lattice Green function, Phil. Trans. Roy. Soc., 273 (1972), 583-610.

J. Wimp, Review of book "A=B" by M. Petkovsek et al., Mathematical Intelligencer, 23 (No. 4, 2001), pp. 72-77.

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

C(2n, n)*Sum_{k=0..n} C(n, k)^2*C(2k, k).

a(n) = (4^n*p(1/2, n)/n!)*hypergeom([ -n, -n, 1/2], [1, 1], 4)), where p(a, k) = product(a+i, i=0..k-1).

E.g.f.: Sum[n>=0, a(n)*x^(2n)] = BesselI(0, 2x)^3.

n^3*a(n) = 2*(2*n-1)*(10*n^2-10*n+3)*a(n-1)-36*(n-1)*(2*n-1)*(2*n-3)*a(n-2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 16 2004

Comment from Cecil C Rousseau (ccrousse(AT)memphis.edu), Mar 14 2006: An asymptotic formula follows immediately from an observation of Bruce Richmond and myself in SIAM Review - 31 (1989, 122-125. We use Hayman's method to find the asymptotic behavior of the sum of squares of the mutinomial coefficients multi(n, k_1, k_2, ...,k_m) with m fixed. From this one gets a_n ~ (3 sqrt(3)/4)*{6^{2n}}/{(pi n)^{3/2}}.

a := proc(n) local k; binomial(2*n, n)*add(binomial(n, k)^2*binomial(2*k, k), k=0..n); end;

C(2n, n) times A002893. Cf. A049020, A049037, A084261.

Cf. 138540.

Sequence in context: A037959 A006480 A138462 this_sequence A004996 A001499 A132467

Adjacent sequences: A002893 A002894 A002895 this_sequence A002897 A002898 A002899

nonn,easy,walk,nice

njas