A185697 Array to be read by rows: Number of ways of placing n rods of length L in a LxLxL simple cubic lattice without any two rods intersecting. (Consecutive rows are for L>=0; in each row, 0<=n<=L^2.)

1, 1, 3, 1, 12, 42, 44, 9, 1, 27, 270, 1278, 3078, 3888, 2592, 918, 189, 21, 1, 48, 936, 9744, 60132, 232560, 585576, 989424, 1151910, 949136, 571608, 261360, 93540, 26160, 5400, 720, 45, 1, 75, 2400, 43400, 496200, 3812640, 20493900, 79452300, 227904675, 494715775, 829703340, 1096775400, 1165877050, 1017460800, 744417900, 465554460, 252490875, 119428125, 49008500, 17175000, 5015100, 1180200, 213900, 27900, 2325, 93

Offset

0,3

Comments

A rod is 'placed' into a LxLxL lattice by occupying a row of L sites in the lattice, along any of the 3 orthogonal directions. Rods cannot intersect; that is, a site cannot belong to more than one rod.

This appears in connection to the degeneracy of the ground states of certain models in lattice statistics.

For each L, there can be at most L^2 rods in the cubic lattice. The terms in the data section cover the range 0<=L<=5. The adjoining b-file contains all terms up to L=16.

The (ordinary) generating function f(L,x) is also formally the partition function for the statistics of the associated lattice model, if one regards the expansion variable x as the 'activity' (the physical interpretation of this parameter depends on the context in which the model appears).

Links

Marcos Simoes, Table of n, a(n) for n = 0..1512

Formula

G.f.: f(L,x) = Sum_{a=0..L} Sum_{b=0..L} Sum_{c=0..L} ( binomial(L,a) * binomial(L,b) * binomial(L,c) * ((1+x)^(L-c)-1)^a * ((1+x)^(L-a)-1)^b * ((1+x)^(L-b)-1)^c ).

Example

The values for L=0,1,2,3 read, respectively:
1
1 3
1 12 42 44 9
1 27 270 1278 3078 3888 2592 918 189 21

Mathematica

Unprotect[Power]; Power[0, 0]=1; Protect[Power]; (* This is necessary so that 0^0=1. *)
(* This defines the generating function: *)
f[L_, x_]:=Sum[Binomial[L, a]*Binomial[L, b]*Binomial[L, c]*((1 + x)^(L - c) - 1)^a*((1 + x)^(L - a) - 1)^b*((1 + x)^(L - b) - 1)^c, {a, 0, L}, {b, 0, L}, {c, 0, L}]
(* The following computes a specific value of the sequence: *)
a[L_, n_]=Coefficient[f[L, x], x, n]
(* The array of coefficients up to L=16 may be generated by: *)
Table[CoefficientList[f[L, x], x], {L, 0, 16}]

Crossrefs

Row sums of this sequence appear in A185702.

Row L=3 gives A181323.

Sequence in context: A186695 A210587 A019232 * A348829 A321697 A263008

Adjacent sequences: A185694 A185695 A185696 * A185698 A185699 A185700

Keyword

nonn,tabf

Author

Marcos Simoes, Feb 10 2011

Status

approved