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A335570 Number A(n,k) of n-step k-dimensional nonnegative lattice walks starting at the origin and using steps that increment all components or decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals. 6
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 7, 6, 1, 1, 1, 5, 13, 17, 10, 1, 1, 1, 6, 21, 40, 47, 20, 1, 1, 1, 7, 31, 81, 136, 125, 35, 1, 1, 1, 8, 43, 146, 325, 496, 333, 70, 1, 1, 1, 9, 57, 241, 686, 1433, 1753, 939, 126, 1, 1, 1, 10, 73, 372, 1315, 3476, 6473, 6256, 2597, 252, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

Alois P. Heinz, Antidiagonals n = 0..60, flattened

EXAMPLE

Square array A(n,k) begins:

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

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

  1,  2,   3,    4,    5,     6,     7,      8, ...

  1,  3,   7,   13,   21,    31,    43,     57, ...

  1,  6,  17,   40,   81,   146,   241,    372, ...

  1, 10,  47,  136,  325,   686,  1315,   2332, ...

  1, 20, 125,  496, 1433,  3476,  7525,  14960, ...

  1, 35, 333, 1753, 6473, 18711, 46165, 102173, ...

MAPLE

b:= proc(n, l) option remember; `if`(n=0, 1, b(n-1, map(x-> x+1, l))+add(

     `if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..nops(l)))

    end:

A:= (n, k)-> b(n, [0$k]):

seq(seq(A(n, d-n), n=0..d), d=0..12);

MATHEMATICA

b[n_, l_] := b[n, l] = If[n == 0, 1, b[n - 1, l + 1] + Sum[If[l[[i]] > 0, b[n - 1, Sort[ReplacePart[l, i -> l[[i]] - 1]]], 0], {i, 1, Length[l]}]];

A[n_, k_] := b[n, Table[0, {k}]];

Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Jan 29 2021, after Alois P. Heinz *)

CROSSREFS

Columns k=0-3 give: A000012, A001405, A151265, A149424.

Rows n=0+1,2-3 give: A000012, A000027(k+1), A002061(k+1).

Main diagonal gives A335588.

Cf. A340591.

Sequence in context: A303912 A133815 A305027 * A323718 A130580 A220708

Adjacent sequences:  A335567 A335568 A335569 * A335571 A335572 A335573

KEYWORD

nonn,tabl,walk

AUTHOR

Alois P. Heinz, Jan 26 2021

STATUS

approved

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Last modified June 19 20:48 EDT 2021. Contains 345145 sequences. (Running on oeis4.)