A271650 - OEIS

%I #17 Apr 06 2024 09:02:18

%S 1,0,40,480,11880,281280,7506400,210268800,6166993000,187069411200,

%T 5833030976640,186014056166400,6044435339896800,199561060892793600,

%U 6679216425794140800,226213441773789550080,7741313040820500484200

%N Number of n-step excursions on the 5-dimensional f.c.c. lattice.

%C a(n) = number of walks in the integer lattice Z^5 starting and ending at the origin, using only the steps of the form (s_1, ..., s_5) with s_1^2 + ... + s_5^2 = 2, i.e., each possible step has precisely two nonzero entries which can be +1 or -1.

%H Christoph Koutschan, <a href="/A271650/b271650.txt">Table of n, a(n) for n = 0..629</a>

%H D. Broadhurst, <a href="http://carmamaths.org/resources/jon/Preprints/Papers/Submitted Papers/4step-walks/walk-broadhurst.pdf">Bessel moments, random walks and Calabi-Yau equations</a>.

%H A. J. Guttmann, <a href="http://dx.doi.org/10.1088/1751-8113/43/30/305205">Lattice Green's functions in all dimensions</a>, J. Phys. A.: Math. Theor. 43 (2010) 305205

%H C. Koutschan, <a href="http://www.koutschan.de/data/fcc/">Computations for higher-dimensional fcc lattices</a>.

%H C. Koutschan, <a href="http://www.koutschan.de/data/fcc1/fcc5_mop.txt">Differential operator annihilating the generating function</a>.

%H C. Koutschan, <a href="http://www.koutschan.de/data/fcc1/fcc5_rec.txt">Recurrence equation</a>.

%H C. Koutschan, <a href="http://arxiv.org/abs/1108.2164">Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices</a>, arXiv:1108.2164 [math.CO], 2011-2013.

%H C. Koutschan, <a href="http://dx.doi.org/10.1088/1751-8113/46/12/125005">Lattice Green's functions of the higher-dimensional face-centered cubic lattices</a>, Journal of Physics A: Mathematical and Theoretical 46(12) (2013), 125005.

%F a(n) satisfies a seventh-order linear recurrence equation with polynomial coefficients of degree 12 (see link above).

%F The probability generating function P(z) = Sum_{n>=0} a(n)*(z/40)^n is given by the 5-fold integral (1/Pi)^5 Int_{0..Pi} ... Int_{0..Pi} 1/(1-z*lambda_5) dk_1 ... dk_5, where the structure function is defined as lambda_5 = (1/binomial(5,2)) Sum_{i=1..5} Sum_{j=(i+1)..5} cos(k_i)*cos(k_j). The function P(z) satisfies a sixth-order linear ODE with polynomial coefficients of degree 17 (see link above).

%e There is one walk with no steps.

%e No walk with a single steps returns to the origin.

%e The number of returning walks with two steps is exactly the number of allowed steps (called the coordination number of the lattice): a(2) = 4*binomial(5,2).

%p nmax := 50: tt := [seq([seq(add(binomial(2*p,p)*binomial(2*j,2*p-n)*binomial(2*n+2*j-2*p,n+j-p), p = floor((n+1)/2)..floor((n+2*j)/2)), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: for d1 from 3 to 5 do tt := [seq([seq(add(binomial(n,p)*add(binomial(2*j,2*q-p)*binomial(2*j+2*p-2*q,j+p-q)*tt[n-p+1,q+1], q = floor((p+1)/2)..floor((p+2*j)/2)), p = 0..n), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: od: [seq(tt[n+1,1], n = 0..nmax)];

%t nmax = 50; T = Table[Sum[Binomial[2 p, p]*Binomial[2 j, 2 p - n]*Binomial[2 n + 2 j - 2 p, n + j - p], {p, Floor[(n + 1)/2], Floor[(n + 2 j)/2]}], {n, 0, nmax}, {j, 0, Floor[(nmax - n)/2]}]; Do[T = Table[Sum[Binomial[n, p]*Sum[Binomial[2 j, 2 q - p]*Binomial[2 j + 2 p - 2 q, j + p - q]*T[[n - p + 1, q + 1]], {q, Floor[(p + 1)/2], Floor[(p + 2 j)/2]}], {p, 0, n}], {n, 0, nmax}, {j, 0, If[d1 < 5, Floor[(nmax - n)/2], 0]}], {d1, 3, 5}]; First /@ T

%Y Cf. A002899 (d = 3, i.e., excursions on the 3-dimensional f.c.c. lattice), A271432 (d = 4), this sequence (d = 5), A271651 (d = 6), A271670 (d = 7), A271671 (d = 8), A271672 (d = 9), A271673 (d = 10), A271674 (d = 11).

%K nonn,walk

%O 0,3

%A _Christoph Koutschan_, Apr 11 2016