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A002896
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Number of 2n-step polygons on cubic lattice.
(Formerly M4285 N1791)
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27
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1, 6, 90, 1860, 44730, 1172556, 32496156, 936369720, 27770358330, 842090474940, 25989269017140, 813689707488840, 25780447171287900, 825043888527957000, 26630804377937061000, 865978374333905289360, 28342398385058078078010, 932905175625150142902300
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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
Diagonal of the rational function R(x,y,z,w) = 1/(1-(w*x*y+w*x*z+w*y+x*z+y+z)). - Gheorghe Coserea, Jul 14 2016
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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).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..645
David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Nachum Dershowitz, Touchard’s Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
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 [math.NT], 2008-2010.
Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.
Chunwei Song, Bowen Yao, On Combinatorial Rectangles with Minimum oo-Discrepancy, arXiv:1909.05648 [math.CO], 2019. See p. 7 for another interpretation.
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
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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
0 = (-x^2+40*x^3-144*x^4)*y''' + (-3*x+180*x^2-864*x^3)*y'' + (-1+132*x-972*x^2)*y' + (6-108*x)*y, where y is the g.f. - Gheorghe Coserea, Jul 14 2016
a(n) = [(x y z)^0] (x+1/x+y+1/y+z+1/z)^(2*n). - Christopher J. Smyth, Sep 25 2018
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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
A002896 := n -> binomial(2*n, n)*hypergeom([1/2, -n, -n], [1, 1], 4):
seq(simplify(A002896(n)), n=0..16); # Peter Luschny, May 23 2017
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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.
Related to diagonal of rational functions: A268545-A268555.
Sequence in context: A201073 A006480 A138462 * A266734 A004996 A001499
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
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STATUS
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approved
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