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A328347
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Number T(n,k) of n-step walks on cubic lattice starting at (0,0,0), ending at (0,k,n-k) and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1); triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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7
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1, 1, 1, 3, 4, 3, 7, 15, 15, 7, 19, 52, 72, 52, 19, 51, 175, 300, 300, 175, 51, 141, 576, 1185, 1480, 1185, 576, 141, 393, 1869, 4473, 6685, 6685, 4473, 1869, 393, 1107, 6000, 16380, 28392, 33880, 28392, 16380, 6000, 1107, 3139, 19107, 58572, 115332, 159264, 159264, 115332, 58572, 19107, 3139
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OFFSET
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0,4
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COMMENTS
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These walks are not restricted to the first (nonnegative) octant.
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LINKS
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FORMULA
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T(n,k) = T(n,n-k).
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EXAMPLE
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Triangle T(n,k) begins:
1;
1, 1;
3, 4, 3;
7, 15, 15, 7;
19, 52, 72, 52, 19;
51, 175, 300, 300, 175, 51;
141, 576, 1185, 1480, 1185, 576, 141;
393, 1869, 4473, 6685, 6685, 4473, 1869, 393;
1107, 6000, 16380, 28392, 33880, 28392, 16380, 6000, 1107;
...
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MAPLE
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b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(add(
add(`if`(i+j+k=1, (h-> `if`(add(t, t=h)<0, 0, b(h)))(
sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
end:
T:= (n, k)-> b(sort([0, k, n-k])):
seq(seq(T(n, k), k=0..n), n=0..12);
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MATHEMATICA
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b[l_List] := b[l] = If[l[[-1]] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[Total[h] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];
T[n_, k_] := b[Sort[{0, k, n - k}]];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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