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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-Fran├žois 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.)