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%I #14 Nov 14 2016 07:21:50
%S 1,1,3,1,12,42,44,9,1,27,270,1278,3078,3888,2592,918,189,21,1,48,936,
%T 9744,60132,232560,585576,989424,1151910,949136,571608,261360,93540,
%U 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
%N 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.)
%C 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.
%C This appears in connection to the degeneracy of the ground states of certain models in lattice statistics.
%C 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.
%C 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).
%H Marcos Simoes, <a href="/A185697/b185697.txt">Table of n, a(n) for n = 0..1512</a>
%F 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 ).
%e The values for L=0,1,2,3 read, respectively:
%e 1
%e 1 3
%e 1 12 42 44 9
%e 1 27 270 1278 3078 3888 2592 918 189 21
%t Unprotect[Power]; Power[0,0]=1; Protect[Power]; (* This is necessary so that 0^0=1. *)
%t (* This defines the generating function: *)
%t 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}]
%t (* The following computes a specific value of the sequence: *)
%t a[L_,n_]=Coefficient[f[L,x],x,n]
%t (* The array of coefficients up to L=16 may be generated by: *)
%t Table[CoefficientList[f[L,x],x],{L,0,16}]
%Y Row sums of this sequence appear in A185702.
%Y Row L=3 gives A181323.
%K nonn,tabf
%O 0,3
%A _Marcos Simoes_, Feb 10 2011
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