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A328300
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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), remaining in the first (nonnegative) octant 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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6
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1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 26, 15, 1, 1, 31, 82, 82, 31, 1, 1, 63, 237, 343, 237, 63, 1, 1, 127, 651, 1257, 1257, 651, 127, 1, 1, 255, 1730, 4256, 5594, 4256, 1730, 255, 1, 1, 511, 4494, 13669, 22411, 22411, 13669, 4494, 511, 1, 1, 1023, 11485, 42279, 83680, 103730, 83680, 42279, 11485, 1023, 1
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OFFSET
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0,5
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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;
1, 3, 1;
1, 7, 7, 1;
1, 15, 26, 15, 1;
1, 31, 82, 82, 31, 1;
1, 63, 237, 343, 237, 63, 1;
1, 127, 651, 1257, 1257, 651, 127, 1;
1, 255, 1730, 4256, 5594, 4256, 1730, 255, 1;
...
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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`(h[1]<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, Function[r, Sum[Sum[Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {k, r}], {j, r}], {i, r}]][{-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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