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A002896
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Number of 2n-step polygons on cubic lattice.
(Formerly M4285 N1791)
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10
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1, 6, 90, 1860, 44730, 1172556, 32496156, 936369720, 27770358330, 842090474940, 25989269017140, 813689707488840, 25780447171287900, 825043888527957000, 26630804377937061000, 865978374333905289360, 28342398385058078078010
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OFFSET
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0,2
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COMMENTS
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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 2n-th moment of tr(A^k) for all k >= 7. - Andrew V. Sutherland, Mar 24 2008
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REFERENCES
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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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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..200
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.
W. K. Hayman, A Generalisation of Stirling's Formula, Journal für die reine und angewandte Mathematik, (1956), vol. 196, pp. 67-95
J. A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, Journal of Statistical Computation and Simulation, 51 (1995), 291-313.
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.
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, arXiv:0803.4462
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FORMULA
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a(n) = 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)/(2n)!] = BesselI(0, 2x)^3. - Corrected by Christopher J. Smyth, Oct 29 2012
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, Jul 16 2004
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 multinomial 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}}. - Cecil C Rousseau (ccrousse(AT)memphis.edu), Mar 14 2006
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
PSUM transform is A174516. - Michael Somos, May 21 2013
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EXAMPLE
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1 + 6*x + 90*x^2 + 1860*x^3 + 44730*x^4 + 1172556*x^5 + 32496156*x^6 + ...
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MAPLE
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a := proc(n) local k; binomial(2*n, n)*add(binomial(n, k)^2 *binomial(2*k, k), k=0..n); end;
# second Maple program
a:= proc(n) option remember; `if`(n<2, 5*n+1,
(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)) /n^3)
end:
seq (a(n), n=0..20); # Alois P. Heinz, Nov 02 2012
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MATHEMATICA
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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, Oct 29 2011 *)
Table[Binomial[2n, n]Sum[Binomial[n, k]^2 Binomial[2k, k], {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Jan 24 2012 *)
a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {-n, -n, 1/2}, {1, 1}, 4] Binomial[ 2 n, n]] (* Michael Somos, May 21 2013 *)
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PROG
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(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
(Sage)
def A002896():
x, y, n = 1, 6, 1
while true:
yield x
n += 1
x, y = y, ((4*n-2)*((10*(n-1)*n+3)*y-18*(n-1)*(2*n-3)*x))/n^3
a = A002896()
[a.next() for i in range(17)] # Peter Luschny, Oct 09 2013
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CROSSREFS
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C(2n, n) times A002893.
Cf. A049020, A049037, A084261, A138540, A174516.
Sequence in context: A201073 A006480 A138462 * A004996 A001499 A147630
Adjacent sequences: A002893 A002894 A002895 * A002897 A002898 A002899
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KEYWORD
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nonn,easy,walk,nice
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AUTHOR
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N. J. A. Sloane, Apr 30 1991
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STATUS
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approved
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