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A229345 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by the same positive integer; square array A(n,k), n>=0, k>=0, read by antidiagonals. 7
1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 7, 22, 4, 1, 1, 25, 248, 188, 8, 1, 1, 121, 6506, 11380, 1712, 16, 1, 1, 721, 292442, 2359348, 577124, 16098, 32, 1, 1, 5041, 19450082, 1088626684, 991365512, 30970588, 154352, 64, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Alois P. Heinz, Antidiagonals n = 0..20, flattened

EXAMPLE

A(2,2) = 22: [(2,2),(1,1),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)], [(2,2),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(1,2),(1,0),(0,0)], [(2,2),(0,2),(0,1),(0,0)], [(2,2),(0,2),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(0,1),(0,0)], [(2,2),(2,1),(2,0),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,0)], [(2,2),(2,0),(1,0),(0,0)], [(2,2),(2,0),(0,0)].

Square array A(n,k) begins:

  1,  1,     1,        1,            1,                 1, ...

  1,  1,     3,        7,           25,               121, ...

  1,  2,    22,      248,         6506,            292442, ...

  1,  4,   188,    11380,      2359348,        1088626684, ...

  1,  8,  1712,   577124,    991365512,     4943064622568, ...

  1, 16, 16098, 30970588, 453530591824, 25162900228200976, ...

MAPLE

b:= proc(l) option remember; local m; m:= nops(l);

      `if`(m=0 or l[m]=0, 1,

      `if`(m>1, add(b(l-[j$m]), j=1..l[1]), 0)+

      add(add(b(sort(subsop(i=l[i]-j, l))), j=1..l[i]), i=1..m))

    end:

A:= (n, k)-> b([n$k]):

seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Sep 24 2013

MATHEMATICA

b[l_] := b[l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[m > 1, Sum[b[l - Array[j&, m]], {j, 1, l[[1]]}],  0] + Sum[Sum[b[Sort[ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, m}]]]; a[n_, k_] := b[Array[n&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-Fran├žois Alcover, Dec 16 2013, translated from Maple *)

CROSSREFS

Columns k=0-3 give: A000012, A011782, A132595(n+1), A229482.

Rows n=0-2 give: A000012, A038507 (for k>1), A229465.

Main diagonal gives: A229346.

Cf. A060854, A227578, A227655, A225094, A210472, A229142, A262809, A263159.

Sequence in context: A078424 A291117 A293181 * A240235 A092742 A316564

Adjacent sequences:  A229342 A229343 A229344 * A229346 A229347 A229348

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 24 2013

STATUS

approved

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Last modified March 18 17:51 EDT 2019. Contains 321292 sequences. (Running on oeis4.)