A271650 Number of n-step excursions on the 5-dimensional f.c.c. lattice.

1, 0, 40, 480, 11880, 281280, 7506400, 210268800, 6166993000, 187069411200, 5833030976640, 186014056166400, 6044435339896800, 199561060892793600, 6679216425794140800, 226213441773789550080, 7741313040820500484200

Offset

0,3

Comments

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.

Links

Christoph Koutschan, Table of n, a(n) for n = 0..629

D. Broadhurst, Bessel moments, random walks and Calabi-Yau equations.

A. J. Guttmann, Lattice Green's functions in all dimensions, J. Phys. A.: Math. Theor. 43 (2010) 305205

C. Koutschan, Computations for higher-dimensional fcc lattices.

C. Koutschan, Differential operator annihilating the generating function.

C. Koutschan, Recurrence equation.

C. Koutschan, Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices, arXiv:1108.2164 [math.CO], 2011-2013.

C. Koutschan, Lattice Green's functions of the higher-dimensional face-centered cubic lattices, Journal of Physics A: Mathematical and Theoretical 46(12) (2013), 125005.

Formula

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

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).

Example

There is one walk with no steps.
No walk with a single steps returns to the origin.
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).

Maple

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)];

Mathematica

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

Crossrefs

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).

Sequence in context: A290611 A115189 A186933 * A177096 A202902 A140045

Adjacent sequences: A271647 A271648 A271649 * A271651 A271652 A271653

Keyword

nonn,walk

Author

Christoph Koutschan, Apr 11 2016

Status

approved