%I A002896 M4285 N1791
%S A002896 1,6,90,1860,44730,1172556,32496156,936369720,27770358330,842090474940,
%T A002896 25989269017140,813689707488840,25780447171287900,825043888527957000,
%U A002896 26630804377937061000,865978374333905289360,28342398385058078078010
%N A002896 Number of 2n-step polygons on cubic lattice.
%C A002896 Number of walks with 2n steps on the cubic lattice Z x Z x Z beginning 
               and ending at (0,0,0)).
%C A002896 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
%D A002896 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002896 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002896 David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, 
               Elliptic integral evaluations of Bessel moments, arXiv:0801.0891.
%D A002896 C. Domb, On the theory of cooperative phenomena in crystals, Advances 
               in Phys., 9 (1960), 149-361.
%D A002896 J. A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, 
               Journal of Statistical Computation and Simulation, 51 (1995), 291-313.
%D A002896 G. S. Joyce, The simple cubic lattice Green function, Phil. Trans. Roy. 
               Soc., 273 (1972), 583-610.
%D A002896 J. Wimp, Review of book "A=B" by M. Petkovsek et al., Mathematical Intelligencer, 
               23 (No. 4, 2001), pp. 72-77.
%D A002896 Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials 
               and random matrices", preprint, 2008.
%F A002896 C(2n, n)*Sum_{k=0..n} C(n, k)^2*C(2k, k).
%F A002896 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).
%F A002896 E.g.f.: Sum[n>=0, a(n)*x^(2n)] = BesselI(0, 2x)^3.
%F A002896 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.rs), Jul 16 2004
%F A002896 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}}.
%p A002896 a := proc(n) local k; binomial(2*n,n)*add(binomial(n,k)^2*binomial(2*k,
               k),k=0..n); end;
%Y A002896 C(2n, n) times A002893. Cf. A049020, A049037, A084261.
%Y A002896 Cf. 138540.
%Y A002896 Sequence in context: A037959 A006480 A138462 this_sequence A004996 A001499 
               A147630
%Y A002896 Adjacent sequences: A002893 A002894 A002895 this_sequence A002897 A002898 
               A002899
%K A002896 nonn,easy,walk,nice
%O A002896 0,2
%A A002896 N. J. A. Sloane (njas(AT)research.att.com).