%I M4285 N1791 #193 Oct 14 2024 12:17:36

%S 1,6,90,1860,44730,1172556,32496156,936369720,27770358330,

%T 842090474940,25989269017140,813689707488840,25780447171287900,

%U 825043888527957000,26630804377937061000,865978374333905289360,28342398385058078078010,932905175625150142902300

%N Number of 2n-step polygons on cubic lattice.

%C Number of walks with 2n steps on the cubic lattice Z^3 beginning and ending at (0,0,0).

%C 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

%C 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

%C Constant term in the expansion of (x + 1/x + y + 1/y + z + 1/z)^(2n). - _Harry Richman_, Apr 29 2020

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A002896/b002896.txt">Table of n, a(n) for n = 0..645</a>

%H David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, <a href="http://arxiv.org/abs/0801.0891">Elliptic integral evaluations of Bessel moments</a>, arXiv:0801.0891 [hep-th], 2008.

%H C. Domb, <a href="http://dx.doi.org/10.1080/00018736000101199">On the theory of cooperative phenomena in crystals</a>, Advances in Phys., 9 (1960), 149-361.

%H Nachum Dershowitz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Dershowitz/dersh3.html">Touchard’s Drunkard</a>, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

%H Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, <a href="https://arxiv.org/abs/2410.07435">The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices</a>, arXiv:2410.07435 [math.CO], 2024. See p. 5.

%H Li Gan, <a href="https://arxiv.org/abs/2307.08732">On the algebraic area of cubic lattice walks</a>, arXiv:2307.08732 [math-ph], 2023.

%H W. K. Hayman, <a href="https://eudml.org/doc/150316">A Generalisation of Stirling's Formula</a>, Journal für die reine und angewandte Mathematik, (1956), vol. 196, pp. 67-95

%H John A. Hendrickson, Jr., <a href="http://dx.doi.org/10.1080/00949659508811639">On the enumeration of rectangular (0,1)-matrices</a>, Journal of Statistical Computation and Simulation, 51 (1995), 291-313.

%H Heng Huat Chan, Yoshio Tanigawa, Yifan Yang, and Wadim Zudilin, <a href="http://dx.doi.org/10.1016/j.aim.2011.06.011">New analogues of Clausen's identities arising from the theory of modular forms</a>, Advances in Mathematics, 228:2 (2011), pp. 1294-1314; doi:10.1016/j.aim.2011.06.011.

%H G. S. Joyce, <a href="http://www.jstor.org/stable/74037">The simple cubic lattice Green function</a>, Phil. Trans. Roy. Soc., 273 (1972), 583-610.

%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</a>, arXiv:0803.4462 [math.NT], 2008-2010.

%H Yen Lee Loh, <a href="https://arxiv.org/abs/1706.03083">A general method for calculating lattice Green functions on the branch cut</a>, arXiv:1706.03083 [math-ph], 2017.

%H Nobu C. Shirai and Naoyuki Sakumichi, <a href="https://arxiv.org/abs/2408.14992">Universal Negative Energetic Elasticity in Polymer Chains: Crossovers among Random, Self-Avoiding, and Neighbor-Avoiding Walks</a>, arXiv:2408.14992 [cond-mat.soft], 2024. See p. 5.

%H Chunwei Song and Bowen Yao, <a href="https://arxiv.org/abs/1909.05648">On Combinatorial Rectangles with Minimum oo-Discrepancy</a>, arXiv:1909.05648 [math.CO], 2019. See p. 7 for another interpretation.

%H Ralph A. Raimi and Jet Wimp, <a href="http://dx.doi.org/10.1007/BF03024606">Review of book "A=B" by M. Petkovsek et al.</a>, Mathematical Intelligencer, 23 (No. 4, 2001), pp. 72-77.

%F a(n) = C(2*n, n)*Sum_{k=0..n} C(n, k)^2*C(2*k, k).

%F 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).

%F 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

%F 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

%F 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

%F 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

%F a(n) = binomial(2*n,n)*A002893(n). - _Mark van Hoeij_, Oct 29 2011

%F 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

%F PSUM transform is A174516. - _Michael Somos_, May 21 2013

%F 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

%F a(n) = [(x y z)^0] (x + 1/x + y + 1/y + z + 1/z)^(2*n). - _Christopher J. Smyth_, Sep 25 2018

%F 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

%F 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

%F 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

%e 1 + 6*x + 90*x^2 + 1860*x^3 + 44730*x^4 + 1172556*x^5 + 32496156*x^6 + ...

%p a := proc(n) local k; binomial(2*n,n)*add(binomial(n,k)^2 *binomial(2*k,k), k=0..n); end;

%p # second Maple program

%p a:= proc(n) option remember; `if`(n<2, 5*n+1,

%p (2*(2*n-1)*(10*n^2-10*n+3) *a(n-1)

%p -36*(n-1)*(2*n-1)*(2*n-3) *a(n-2)) /n^3)

%p end:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Nov 02 2012

%p A002896 := n -> binomial(2*n,n)*hypergeom([1/2, -n, -n], [1, 1], 4):

%p seq(simplify(A002896(n)), n=0..16); # _Peter Luschny_, May 23 2017

%t 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 *)

%t Table[Binomial[2n,n]Sum[Binomial[n,k]^2 Binomial[2k,k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Jan 24 2012 *)

%t a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {-n, -n, 1/2}, {1, 1}, 4] Binomial[ 2 n, n]] (* _Michael Somos_, May 21 2013 *)

%o (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

%o (Sage)

%o def A002896():

%o x, y, n = 1, 6, 1

%o while True:

%o yield x

%o n += 1

%o x, y = y, ((4*n-2)*((10*(n-1)*n+3)*y-18*(n-1)*(2*n-3)*x))//n^3

%o a = A002896()

%o [next(a) for i in range(17)] # _Peter Luschny_, Oct 09 2013

%Y C(2n, n) times A002893.

%Y Cf. A049020, A049037, A084261, A138540, A174516.

%Y Related to diagonal of rational functions: A268545-A268555.

%Y Row k=3 of A287318.

%K nonn,easy,walk,nice

%O 0,2

%A _N. J. A. Sloane_