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%I #14 Feb 28 2022 07:48:16
%S 1,8,0,27,24,0,64,108,48,0,125,288,342,96,0,216,600,1056,1104,144,0,
%T 343,1080,2370,3984,3240,240,0,512,1764,4464,9612,14256,9504,192,0,
%U 729,2688,7518,18888,37470,51504,25344,144,0,1000,3888,11712,32712,77184,148224
%N T(n,k) = Number of n-step self-avoiding walks on a k X k X k cube summed over all starting positions.
%H R. H. Hardin, <a href="/A187162/b187162.txt">Table of n, a(n) for n = 1..241</a>
%F a(1,k) = k^3
%F a(2,k) = 6*k^3 - 6*k^2
%F a(3,k) = 30*k^3 - 60*k^2 + 24*k for k>1
%F a(4,k) = 150*k^3 - 426*k^2 + 312*k - 48 for k>2
%F a(5,k) = 726*k^3 - 2640*k^2 + 2688*k - 720 for k>3
%F a(6,k) = 3534*k^3 - 15366*k^2 + 19536*k - 7056 for k>4
%F a(7,k) = 16926*k^3 - 85380*k^2 + 128832*k - 57312 for k>5
%F a(8,k) = 81390*k^3 - 463074*k^2 + 801216*k - 418032 for k>6
%F a(9,k) = 387966*k^3 - 2452704*k^2 + 4766544*k - 2833872 for k>7
%F a(10,k) = 1853886*k^3 - 12825630*k^2 + 27515184*k - 18252624 for k>8
%F ["Empirical" removed by _Andrey Zabolotskiy_, Feb 28 2022]
%e Solution for n=9 3X3X3
%e 0 0 0 9 0 0 8 0 0
%e 0 2 1 6 0 0 7 0 0
%e 0 3 0 5 4 0 0 0 0
%e Table starts
%e 1 8 27 64 125 216 343 512 729 1000
%e 0 24 108 288 600 1080 1764 2688 3888 5400
%e 0 48 342 1056 2370 4464 7518 11712 17226 24240
%e 0 96 1104 3984 9612 18888 32712 51984 77604 110472
%e 0 144 3240 14256 37470 77184 137754 223536 338886 488160
%e 0 240 9504 51504 148224 320328 588924 975216 1500408 2185704
%e 0 192 25344 177120 568248 1298016 2466510 4175136 6525450 9619008
%e 0 144 67824 608928 2188608 5299056 10416624 18026640 28617228 42676728
%e 0 0 167016 2013360 8227752 21274896 43422072 76964016 124223214 187527168
%e 0 0 414912 6654048 30938640 85654320 181790352 330218544 541990896 828222216
%Y Cf. A187163, A187164, A187165, A187166, A187167, A187168, A187169, A187170, A187171.
%K nonn,tabl
%O 1,2
%A _R. H. Hardin_, Mar 06 2011
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