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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
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-François 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.
Sequence in context: A078424 A291117 A293181 * A240235 A092742 A316564
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 24 2013
STATUS
approved

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Last modified May 28 19:55 EDT 2024. Contains 372919 sequences. (Running on oeis4.)