OFFSET
0,2
COMMENTS
Number of walks with 2n steps on the cubic lattice Z^3 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
Constant term in the expansion of (x + 1/x + y + 1/y + z + 1/z)^(2n). - Harry Richman, Apr 29 2020
REFERENCES
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).
LINKS
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.
Nikolai Beluhov, Powers of 2 in Balanced Grid Colourings, arXiv:2504.21451 [math.CO], 2025.
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.
Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024. See p. 5.
Li Gan, On the algebraic area of cubic lattice walks, arXiv:2307.08732 [math-ph], 2023.
S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 23.
W. K. Hayman, A Generalisation of Stirling's Formula, Journal für die reine und angewandte Mathematik, (1956), vol. 196, pp. 67-95.
John 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, 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.
Ralph A. Raimi and Jet Wimp, Review of book "A=B" by M. Petkovsek et al., Mathematical Intelligencer, 23 (No. 4, 2001), pp. 72-77.
Nobu C. Shirai and Naoyuki Sakumichi, Universal Negative Energetic Elasticity in Polymer Chains: Crossovers among Random, Self-Avoiding, and Neighbor-Avoiding Walks, arXiv:2408.14992 [cond-mat.soft], 2024. See p. 5.
Chunwei Song and Bowen Yao, On Combinatorial Rectangles with Minimum oo-Discrepancy, arXiv:1909.05648 [math.CO], 2019. See p. 7 for another interpretation.
FORMULA
a(n) = C(2*n, n)*Sum_{k=0..n} C(n, k)^2*C(2*k, k).
a(n) = (4^n*p(1/2, n)/n!)*hypergeom([-n, -n, 1/2], [1, 1], 4), where p(a, k) = Product_{i=0..k-1} (a+i).
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2*x)^3. - Corrected by Christopher J. Smyth, Oct 29 2012
D-finite with recurrence: 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 me 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 ~ (3/4)*sqrt(3)*6^(2*n)/(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
a(n) = (1/Pi)^3*Integral_{0 <= x, y, z <= Pi} (2*cos(x) + 2*cos(y) + 2*cos(z))^(2*n) dx dy dz. - Peter Bala, Feb 10 2022
a(n) = Sum_{i+j+k=n, 0<=i,j,k<=n} multinomial(2n [i,i,j,j,k,k]). - Shel Kaphan, Jan 16 2023
Sum_{k>=0} a(k)/36^k = A086231 = (sqrt(3)-1) * (Gamma(1/24) * Gamma(11/24))^2 / (32*Pi^3). - Vaclav Kotesovec, Apr 23 2023
G.f.: HeunG(1/9,1/12,1/4,3/4,1,1/2,4*x)^2 (see Hassani et al.). - Stefano Spezia, Feb 16 2025
EXAMPLE
1 + 6*x + 90*x^2 + 1860*x^3 + 44730*x^4 + 1172556*x^5 + 32496156*x^6 + ...
MAPLE
a := proc(n) local k; binomial(2*n, n)*add(binomial(n, k)^2 *binomial(2*k, k), k=0..n); end;
# Alternative:
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
MATHEMATICA
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 *)
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
(SageMath)
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()
[next(a) for i in range(17)] # Peter Luschny, Oct 09 2013
CROSSREFS
KEYWORD
nonn,easy,walk,nice
AUTHOR
STATUS
approved