Revision History for A271670
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#11 by Peter Luschny at Wed Apr 13 09:09:43 EDT 2016
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#10 by Michel Marcus at Wed Apr 13 05:43:43 EDT 2016
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#9 by Christoph Koutschan at Wed Apr 13 05:37:23 EDT 2016
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#8 by Christoph Koutschan at Wed Apr 13 05:37:05 EDT 2016
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| FORMULA
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The probability generating function P(z) = sumSum_{n>=0} a(n)*(z/84)^n is given by the 7-fold integral (1/Pi)^7 Int_{0..Pi} ... Int_{0..Pi} 1/(1-z*lambda_7) dk_1 ... dk_7, where the structure function is defined as lambda_7 = (1/binomial(7,2)) Sum_{i=1..7} Sum_{j=(i+1)..7} cos(k_i)*cos(k_j). The function P(z) conjecturally satisfies an eleventh-order linear ODE with polynomial coefficients of degree 68 (see link above).
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| CROSSREFS
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Cf. A002899 (d = 3, i.e., excursions on the 3-dimensional f.c.c. lattice), A271432, (d = 4), A271650, (d = 5), A271651. (d = 6), this sequence (d = 7), A271671 (d = 8), A271672 (d = 9), A271673 (d = 10), A271674 (d = 11).
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| STATUS
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approved
editing
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#7 by Alois P. Heinz at Tue Apr 12 12:36:59 EDT 2016
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#6 by Michel Marcus at Tue Apr 12 07:57:38 EDT 2016
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#5 by Michel Marcus at Tue Apr 12 07:57:33 EDT 2016
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#4 by Christoph Koutschan at Tue Apr 12 07:26:06 EDT 2016
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#3 by Christoph Koutschan at Tue Apr 12 07:25:09 EDT 2016
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| LINKS
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Christoph Koutschan, <a href="/A271670/b271670.txt">Table of n, a(n) for n = 0..524</a>
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#2 by Christoph Koutschan at Tue Apr 12 07:23:50 EDT 2016
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| NAME
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allocatedNumber of n-step excursions on forthe Christoph7-dimensional Koutschanf.c.c. lattice.
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| DATA
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1, 0, 84, 1680, 66276, 2731680, 128704800, 6555265920, 355588928100, 20247799145280, 1198746727590384, 73266532153214400, 4598338364703822816, 295145004688715301120, 19311431876483926443264
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| OFFSET
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0,3
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| COMMENTS
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a(n) = number of walks in the integer lattice Z^7 starting and ending at the origin, using only the steps of the form (s_1, ..., s_7) with s_1^2 + ... + s_7^2 = 2, i.e., each possible step has precisely two nonzero entries which can be +1 or -1.
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| LINKS
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C. Koutschan, <a href="http://www.koutschan.de/data/fcc1/">Computations for higher-dimensional fcc lattices</a>.
C. Koutschan, <a href="http://www.koutschan.de/data/fcc1/fcc7_mop.txt">Differential operator annihilating the generating function</a>.
C. Koutschan, <a href="http://www.koutschan.de/data/fcc1/fcc7_rec.txt">Recurrence equation</a>.
N. Zenine, S. Hassani, J-M. Maillard, <a href="http://arxiv.org/abs/1409.8615">Lattice Green Functions: the seven-dimensional face-centred cubic lattice</a>, arXiv:1409.8615 [math-ph], 2014.
N. Zenine, S. Hassani, J-M. Maillard, <a href="http://dx.doi.org/10.1088/1751-8113/49/16/164003">Lattice Green Functions: the seven-dimensional face-centred cubic lattice</a>, Journal of Physics A: Mathematical and Theoretical 48 (2015), 035205.
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| FORMULA
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a(n) conjecturally satisfies a linear recurrence equation of order 15 with polynomial coefficients of degree 56 (see link above).
The probability generating function P(z) = sum_{n>=0} a(n)*(z/84)^n conjecturally satisfies an eleventh-order linear ODE with polynomial coefficients of degree 68 (see link above).
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| EXAMPLE
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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(7,2).
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| MAPLE
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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 7 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)];
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| MATHEMATICA
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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 < 7, Floor[(nmax - n)/2], 0]}], {d1, 3, 7}]; First /@ T
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| CROSSREFS
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Cf. A002899, A271432, A271650, A271651
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| KEYWORD
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allocated
nonn,walk
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| AUTHOR
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Christoph Koutschan, Apr 12 2016
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| STATUS
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approved
editing
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