A306813 -id:A306813 - cp's OEIS frontend

A306814 Number T(n,k) of n-step paths from (0,0) to (0,k) that stay in the first quadrant (but may touch the axes) consisting of steps (-1,0), (0,1), (0,-1) and (1,-1); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 7, 5, 6, 4, 0, 1, 10, 23, 9, 10, 5, 0, 1, 38, 35, 51, 14, 15, 6, 0, 1, 89, 131, 84, 94, 20, 21, 7, 0, 1, 229, 355, 309, 168, 155, 27, 28, 8, 0, 1, 752, 874, 947, 608, 300, 237, 35, 36, 9, 0, 1, 1873, 3081, 2292, 2075, 1070, 495, 343, 44, 45, 10, 0, 1

Offset: 0

Alois P. Heinz, Mar 11 2019

			T(4,2) = 3:
  [(0,0), (0,1), (0,0), (0,1), (0,2)],
  [(0,0), (0,1), (0,2), (0,1), (0,2)],
  [(0,0), (0,1), (0,2), (0,3), (0,2)].
Triangle T(n,k) begins:
    1;
    0,   1;
    1,   0,   1;
    1,   2,   0,   1;
    2,   3,   3,   0,   1;
    7,   5,   6,   4,   0,   1;
   10,  23,   9,  10,   5,   0,  1;
   38,  35,  51,  14,  15,   6,  0,  1;
   89, 131,  84,  94,  20,  21,  7,  0, 1;
  229, 355, 309, 168, 155,  27, 28,  8, 0, 1;
  752, 874, 947, 608, 300, 237, 35, 36, 9, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Column k=0 gives A151346.
Row sums give A151404.
T(2n,n) gives A306813.
T(n+1,n-1) gives A001477.
T(n+2,n-1) gives A000217.
T(n+3,n-1) gives A000096.
Cf. A199915.

b:= proc(n, x, y) option remember; `if`(min(n, x, y, n-x-y)<0, 0,
      `if`(n=0, 1, add(b(n-1, x-d[1], y-d[2]),
       d=[[-1, 0], [0, 1], [0, -1], [1, -1]])))
    end:
T:= (n, k)-> b(n, 0, k):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, x_, y_] := b[n, x, y] = If[Min[n, x, y, n - x - y] < 0, 0, If[n == 0, 1, Sum[b[n - 1, x - d[[1]], y - d[[2]]], {d, {{-1, 0}, {0, 1}, {0, -1}, {1, -1}}}]]];
T[n_, k_] := b[n, 0, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 14 2020, after Maple *)

A317782 Number of 2n-step paths from (0,0) to (0,n) that stay in the first quadrant (but may touch the axes) consisting of steps (1,0), (0,1), (0,-1) and (-1,1).

Original entry on oeis.org

1, 1, 5, 51, 474, 4329, 43406, 469565, 5228459, 59259957, 686003702, 8097484169, 97005128492, 1175916181703, 14404685872773, 178105648065109, 2220134252592683, 27872257776993240, 352143374331177766, 4474477933645201621, 57147423819800882972

Offset: 0

Alois P. Heinz, Sep 24 2018

			a(2) = 5: [(0,1),(0,-1),(0,1),(0,1)], [(0,1),(0,1),(0,-1),(0,1)], [(0,1),(0,1),(0,1),(0,-1)], [(1,0),(-1,1),(1,0),(-1,1)], [(1,0),(1,0),(-1,1),(-1,1)].

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A199915, A306813.

b:= proc(n, x, y) option remember; `if`(min(args, n-x-y)<0, 0, `if`(n=0, 1,
      add(b(n-1, x-d[1], y-d[2]), d=[[1, 0], [0, 1], [0, -1], [-1, 1]])))
    end:
a:= n-> b(2*n, 0, n):
seq(a(n), n=0..25);

b[n_, x_, y_] := b[n, x, y] = If[Min[n, x, y, n - x - y] < 0, 0, If[n == 0, 1, Sum[b[n - 1, x - d[[1]], y - d[[2]]], {d, {{1, 0}, {0, 1}, {0, -1}, {-1, 1}}}]]];
a[n_] := b[2n, 0, n];
a /@ Range[0, 25] (* Jean-François Alcover, May 13 2020, after Maple *)

a(n) = A199915(2n,n).

a(n) ~ c * d^n / n^2, where d = (2 + 4/3^(3/4))^2 = 14.0982628380912972017512943055944... and c = 0.25546328221900708410379626465... - Vaclav Kotesovec, Mar 13 2019, updated Mar 17 2024

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A306814 Number T(n,k) of n-step paths from (0,0) to (0,k) that stay in the first quadrant (but may touch the axes) consisting of steps (-1,0), (0,1), (0,-1) and (1,-1); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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