Revision History for A263159
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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.
(history;
published version)
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#16 by Susanna Cuyler at Sat Apr 25 14:47:42 EDT 2020
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#15 by Jean-François Alcover at Sat Apr 25 12:36:57 EDT 2020
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#14 by Jean-François Alcover at Sat Apr 25 12:36:51 EDT 2020
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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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#13 by Alois P. Heinz at Wed May 09 09:53:37 EDT 2018
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#12 by Alois P. Heinz at Wed May 09 09:53:35 EDT 2018
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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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#11 by Alois P. Heinz at Thu Oct 15 06:36:33 EDT 2015
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#10 by Alois P. Heinz at Thu Oct 15 06:36:28 EDT 2015
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| LINKS
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Alois P. Heinz, <a href="/A263159/b263159.txt">Antidiagonals n = 0..20, flattened</a>
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#9 by Alois P. Heinz at Wed Oct 14 12:05:22 EDT 2015
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#8 by Alois P. Heinz at Wed Oct 14 12:05:07 EDT 2015
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| NAME
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Number A(n,k) of lattice paths starting at {n}^k and ending when k=0 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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#7 by Alois P. Heinz at Sun Oct 11 13:11:34 EDT 2015
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| NAME
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Number A(n,k) of lattice paths starting at {n}^k and ending when k=0 or any component isequals 0 and , 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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