A263159 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.
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
Offset: 0
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..20, flattened
Crossrefs
Programs
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Maple
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); -
Mathematica
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 *)