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A002896
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Number of 2n-step polygons on cubic lattice.
(Formerly M4285 N1791)
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8
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1, 6, 90, 1860, 44730, 1172556, 32496156, 936369720, 27770358330, 842090474940, 25989269017140, 813689707488840, 25780447171287900, 825043888527957000, 26630804377937061000, 865978374333905289360, 28342398385058078078010
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
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REFERENCES
| David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891.
Heng Huat Chan,Yoshio Tanigawa, Yifan Yang, and Wadim Zudilin, "New analogues of Clausen's identities arising from the theory of modular forms", Advances in Mathematics, 228:2 (2011), pp. 1294-1314; doi:10.1016/j.aim.2011.06.011.
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.
Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Wimp, Review of book "A=B" by M. Petkovsek et al., Mathematical Intelligencer, 23 (No. 4, 2001), pp. 72-77.
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FORMULA
| 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.rs), 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}}.
G.f. (1/sqrt(1+12*z)) * hypergeom([1/8,3/8],[1],64/81*z*(1+sqrt(1-36*z))^2*(2+sqrt(1-36*z))^4/(1+12*z)^4) * hypergeom([1/8, 3/8],[1],64/81*z*(1-sqrt(1-36*z))^2*(2-sqrt(1-36*z))^4/(1+12*z)^4). - Sergey Perepechko, Jan 26 2011
a(n) = binomial(2*n,n)*A002893(n). - Mark van Hoeij, Oct 29 2011
G.f. (1/2)*(10-72*x-6*(144*x^2-40*x+1)^(1/2))^(1/2)*hypergeom([1/6, 1/3],[1],54*x*(108*x^2-27*x+1+(9*x-1)*(144*x^2-40*x+1)^(1/2)))^2 - Mark van Hoeij, Nov 12 2011
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MAPLE
| a := proc(n) local k; binomial(2*n, n)*add(binomial(n, k)^2*binomial(2*k, k), k=0..n); end;
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MATHEMATICA
| f[n_] := 4^n*Gamma[n + 1/2]*Sum[Binomial[n, k]^2 Binomial[2 k, k], {k, 0, n}]/(Sqrt[Pi]*n!); Array[f, 17, 0] (* Robert G. Wilson v, 29 Oct 2011 *)
Table[Binomial[2n, n]Sum[Binomial[n, k]^2 Binomial[2k, k], {k, 0, n}], {n, 0, 20}] (* From Harvey P. Dale, Jan 24 2012 *)
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PROG
| (PARI) a(n)=binomial(2*n, n)*sum(k=0, n, binomial(n, k)^2*binomial(2*k, k)) \\ Charles R Greathouse IV, Oct 31 2011
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CROSSREFS
| C(2n, n) times A002893. Cf. A049020, A049037, A084261.
Cf. 138540.
Sequence in context: A037959 A006480 A138462 * A004996 A001499 A147630
Adjacent sequences: A002893 A002894 A002895 * A002897 A002898 A002899
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KEYWORD
| nonn,easy,walk,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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