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A263159
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Number A(n,k) of lattice paths starting at {n}^k and ending when k or any component equals 0, using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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19
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1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 13, 1, 1, 1, 15, 157, 63, 1, 1, 1, 31, 2101, 5419, 321, 1, 1, 1, 63, 32461, 717795, 220561, 1683, 1, 1, 1, 127, 580693, 142090291, 328504401, 9763807, 8989, 1, 1, 1, 255, 11917837, 39991899123, 944362553521, 172924236255, 454635973, 48639, 1, 1
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OFFSET
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0,8
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LINKS
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 3, 7, 15, 31, ...
1, 1, 13, 157, 2101, 32461, ...
1, 1, 63, 5419, 717795, 142090291, ...
1, 1, 321, 220561, 328504401, 944362553521, ...
1, 1, 1683, 9763807, 172924236255, 7622403922836151, ...
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MAPLE
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s:= proc(n) option remember; `if`(n=0, {[]},
map(x-> [[x[], 0], [x[], 1]][], s(n-1)))
end:
b:= proc(l) option remember; `if`(l=[] or l[1]=0, 1,
add((p-> `if`(p[1]<0, 0, `if`(p[1]=0, 1, b(p)))
)(sort(l-x)), x=s(nops(l)) minus {[0$nops(l)]}))
end:
A:= (n, k)-> b([n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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g[k_] := Table[Reverse[IntegerDigits[n, 2]][[;; k]], {n, 2^k+1, 2^(k+1)-1}];
b[l_] := b[l] = If[l[[1]] == 0, 1, Sum[b[Sort[l - h]], {h, g[k]}]];
a[n_, k_] := If[n == 0 || k == 0 || k == 1, 1, b[Table[n, {k}]]];
Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 25 2020, after Alois P. Heinz in A115866 *)
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CROSSREFS
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Columns k=0+1, 2-10 give: A000012, A001850, A115866, A263162, A263163, A263164, A263165, A263166, A263167, A263168.
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KEYWORD
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AUTHOR
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STATUS
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approved
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