A151265 -id:A151265 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Jan 26 2021

			A(2,2) = 3: [(0,0),(1,1),(2,2)], [(0,0),(1,1),(0,1)], [(0,0),(1,1),(1,0)].
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, ...
  ...

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

Columns k=0-10 give: A000012, A001405, A151265, A149424, A346225, A346226, A346227, A346228, A346229, A346230, A346231.
Rows n=0+1,2-3 give: A000012, A000027(k+1), A002061(k+1).
Main diagonal gives A335588.
Cf. A340591.

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);

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-François Alcover, Jan 29 2021, after Alois P. Heinz *)

A(n,k) == 1 (mod k) for k >= 2.

A149424 Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (0, -1, 0), (0, 0, -1), (1, 1, 1)}.

Original entry on oeis.org

1, 1, 4, 13, 40, 136, 496, 1753, 6256, 22912, 85216, 314836, 1170688, 4396048, 16623328, 62744017, 237680992, 904962400, 3459831424, 13219219972, 50621972224, 194465172304, 749061374848, 2884682636764, 11126422372864, 43007603099296, 166555051934848, 644984620465264, 2500560314630656

Offset: 0

Manuel Kauers, Nov 18 2008

Alois P. Heinz, Table of n, a(n) for n = 0..300
A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.

Cf. A151265.

Column k=3 of A335570.

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..3))
    end:
a:= n-> b(n, [0$3]):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 26 2021

aux[i_Integer, j_Integer, k_Integer, n_Integer] := Which[Min[i, j, k, n] < 0 || Max[i, j, k] > n, 0, n == 0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n] = aux[-1 + i, -1 + j, -1 + k, -1 + n] + aux[i, j, 1 + k, -1 + n] + aux[i, 1 + j, k, -1 + n] + aux[1 + i, j, k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]

a(n) == 1 (mod 3). - Alois P. Heinz, Jul 12 2021

cp's OEIS Frontend

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.

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