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A227655
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Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_{i}-p_{i+1}) <= 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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24
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 4, 1, 1, 1, 24, 44, 8, 1, 1, 1, 120, 896, 320, 16, 1, 1, 1, 720, 29392, 33904, 2328, 32, 1, 1, 1, 5040, 1413792, 7453320, 1281696, 16936, 64, 1, 1, 1, 40320, 93770800, 2940381648, 1897242448, 48447504, 123208, 128, 1, 1
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OFFSET
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0,8
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LINKS
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Alois P. Heinz, Antidiagonals n = 0..18, flattened
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EXAMPLE
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A(2,2) = 2^2 = 4:
(1,2) (0,1)
/ \ / \
(2,2) (1,1) (0,0)
\ / \ /
(2,1) (1,0)
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 2, 6, 24, 120, ...
1, 1, 4, 44, 896, 29392, ...
1, 1, 8, 320, 33904, 7453320, ...
1, 1, 16, 2328, 1281696, 1897242448, ...
1, 1, 32, 16936, 48447504, 482913033152, ...
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MAPLE
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b:= proc(l) option remember; `if`({l[]}={0}, 1, add(
`if`(l[i]=0 or i>1 and 1<abs(l[i-1]-l[i]+1) or
i<nops(l) and 1<abs(l[i+1]-l[i]+1), 0,
b(subsop(i=l[i]-1, l))), i=1..nops(l)))
end:
A:= (n, k)-> `if`(k<2, 1, b([n$k])):
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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b[l_] := b[l] = If[Union[l] == {0}, 1, Sum[If[l[[i]] == 0 || i>1 && 1 < Abs[l[[i-1]] - l[[i]] + 1] || i<Length[l] && 1<Abs[l[[i+1]] - l[[i]] + 1], 0, b[ReplacePart[l, i -> l[[i]]-1]]], {i, 1, Length[l]}]]; a[n_, k_] := If[k<2, 1, b[Array[n&, k]]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)
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CROSSREFS
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Columns k=0+1, 2-10 give: A000012, A000079, A227665, A227666, A227667, A227668, A227669, A227670, A227671, A227672.
Rows n=0-10 give: A000012, A000142, A227656, A227657, A227658, A227659, A227660, A227661, A227662, A227663, A227664.
Main diagonal gives A227673.
Cf. A262809, A263159, A318191.
Sequence in context: A213275 A069777 A225816 * A064992 A187783 A089759
Adjacent sequences: A227652 A227653 A227654 * A227656 A227657 A227658
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Jul 19 2013
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STATUS
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approved
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