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 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. 19
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Antidiagonals n = 0..20, flattened EXAMPLE 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, ... 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 *) CROSSREFS Columns k=0+1, 2-10 give: A000012, A001850, A115866, A263162, A263163, A263164, A263165, A263166, A263167, A263168. Rows n=0-1 give: A000012, A255047. Main diagonal gives A263160. Cf. A210472, A225094, A227578, A227655, A229142, A229345, A262809. Sequence in context: A328718 A005765 A343717 * A229142 A156535 A327564 Adjacent sequences:  A263156 A263157 A263158 * A263160 A263161 A263162 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Oct 11 2015 STATUS approved

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Last modified June 13 20:25 EDT 2021. Contains 345009 sequences. (Running on oeis4.)