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A271651 Number of n-step excursions on the 6-dimensional f.c.c. lattice. 11
1, 0, 60, 960, 30780, 996480, 36560400, 1430553600, 59089923900, 2543035488000, 113129280527760, 5170796720812800, 241741903350301200, 11520044551208793600, 558061378022616811200, 27421336248833005839360 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) = number of walks in the integer lattice Z^6 starting and ending at the origin, using only the steps of the form (s_1, ..., s_6) with s_1^2 + ... + s_6^2 = 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..567

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 twelfth-order linear recurrence equation with polynomial coefficients of degree 33 (see link above).

The probability generating function P(z) = Sum_{n>=0} a(n)*(z/60)^n is given by the 6-fold integral (1/Pi)^6 Int_{0..Pi} ... Int_{0..Pi} 1/(1-z*lambda_6) dk_1 ... dk_6, where the structure function is defined as lambda_6 = (1/binomial(6,2)) Sum_{i=1..6} Sum_{j=(i+1)..6} cos(k_i)*cos(k_j). The function P(z) satisfies an eighth-order linear ODE with polynomial coefficients of degree 43 (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(6,2).

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 6 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 < 6, Floor[(nmax - n)/2], 0]}], {d1, 3, 6}]; First /@ T

CROSSREFS

Cf. A002899 (d = 3, i.e., excursions on the 3-dimensional f.c.c. lattice), A271432 (d = 4), A271650 (d = 5), this sequence (d = 6), A271670 (d = 7), A271671 (d = 8), A271672 (d = 9), A271673 (d = 10), A271674 (d = 11).

Sequence in context: A268967 A189607 A223213 * A223348 A305545 A056352

Adjacent sequences:  A271648 A271649 A271650 * A271652 A271653 A271654

KEYWORD

nonn,walk

AUTHOR

Christoph Koutschan, Apr 11 2016

STATUS

approved

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Last modified August 12 16:10 EDT 2020. Contains 336439 sequences. (Running on oeis4.)