[seqfan] Self-avoiding walks on nXnXn cubic lattice, guess the coefficients

[Ron Hardin]

Number of k-step self-avoiding walks on a nXnXn cubic lattice summed over all 
starting positions

T(k,n) Table starts
.1...8.....27.....64....125....216....343...512...729.1000.1331
.0..24....108....288....600...1080...1764..2688..3888.5400.....
.0..48....342...1056...2370...4464...7518.11712.17226..........
.0..96...1104...3984...9612..18888..32712.51984................
.0.144...3240..14256..37470..77184.137754......................
.0.240...9504..51504.148224.320328.............................
.0.192..25344.177120.568248....................................
.0.144..67824.608928...........................................
.0...0.167016..................................................
.0...0.........................................................

Empirical formulas for rows
k=1: a(n) = n^3
k=2: a(n) = 6*n^3 - 6*n^2
k=3: a(n) = 30*n^3 - 60*n^2 + 24*n for n>1
k=4: a(n) = 150*n^3 - 426*n^2 + 312*n - 48 for n>2
k=5: a(n) = 726*n^3 - 2640*n^2 + 2688*n - 720 for n>3
k=6: a(n) = 3534*n^3 - 15366*n^2 + 19536*n - 7056 for n>4
k=7: a(n) = 16926*n^3 - 85380*n^2 + 128832*n - 57312 for n>5
k=8: a(n) = 81390*n^3 - 463074*n^2 + 801216*n - 418032 for n>6

The coefficient of n^3 is http://oeis.org/A001412 Number of n-step self-avoiding 
walks on cubic lattice

rhhardin at mindspring.com
rhhardin at att.net (either)