A317782 -id:A317782 - cp's OEIS frontend

A199915 Triangle read by rows: T(n,k) is the number 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).

1, 0, 1, 1, 1, 1, 1, 2, 3, 1, 2, 7, 5, 6, 1, 7, 10, 21, 14, 10, 1, 10, 38, 48, 51, 35, 15, 1, 38, 89, 135, 168, 120, 76, 21, 1, 89, 229, 441, 458, 474, 281, 147, 28, 1, 229, 752, 1121, 1604, 1475, 1188, 637, 260, 36, 1, 752, 1873, 3692, 4772, 5100, 4329, 2800, 1366, 429, 45, 1

Offset: 0

Alois P. Heinz, Nov 11 2011

			T(4,2) = 5: ((1,0),(1,0),(-1,1),(-1,1)); ((1,0),(-1,1),(1,0),(-1,1)); ((0,1),(0,1),(0,1),(0,-1)); ((0,1),(0,1),(0,-1),(0,1)); ((0,1),(0,-1),(0,1),(0,1)).
Triangle begins:
   1;
   0,  1;
   1,  1,  1;
   1,  2,  3,  1;
   2,  7,  5,  6,  1;
   7, 10, 21, 14, 10,  1;
  10, 38, 48, 51, 35, 15,  1;

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A151346 (columns k=0, 1), A000217(n) = T(n+1,n), A151412 (row sums).
T(2n,n) gives A317782.
Cf. A306814.

b:= proc(n, k, x, y) option remember;
     `if`(n<0 or x<0 or y<0 or n b(n, k, 0, 0):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, k_, x_, y_] := b[n, k, x, y] = If[n<0 || x<0 || y<0 || nJean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

A306813 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, 0, 3, 10, 20, 237, 770, 3944, 28635, 112360, 744084, 4381083, 21579779, 143815322, 801165187, 4578481584, 29176623983, 165772480380, 1013147794546, 6259309820475, 36974951346176, 230752749518819, 1413352914731005, 8618746801792237, 53986291171211635

Offset: 0

Alois P. Heinz, Mar 11 2019

			a(0) = 1: [(0,0)].
a(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)].
a(3) = 10:
  [(0,0), (0,1), (1,0), (1,1), (1,2), (1,3), (0,3)],
  [(0,0), (0,1), (0,2), (1,1), (1,2), (1,3), (0,3)],
  [(0,0), (0,1), (0,2), (0,3), (1,2), (1,3), (0,3)],
  [(0,0), (0,1), (0,2), (0,3), (0,4), (1,3), (0,3)],
  [(0,0), (0,1), (1,0), (1,1), (1,2), (0,2), (0,3)],
  [(0,0), (0,1), (0,2), (1,1), (1,2), (0,2), (0,3)],
  [(0,0), (0,1), (0,2), (0,3), (1,2), (0,2), (0,3)],
  [(0,0), (0,1), (1,0), (1,1), (0,1), (0,2), (0,3)],
  [(0,0), (0,1), (0,2), (1,1), (0,1), (0,2), (0,3)],
  [(0,0), (0,1), (1,0), (0,0), (0,1), (0,2), (0,3)].

Vaclav Kotesovec, Table of n, a(n) for n = 0..1116 (terms 0..500 from Alois P. Heinz)

Cf. A000108, A126596, A174687, A306814, A317782.

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:
a:= n-> b(2*n, 0, n):
seq(a(n), n=0..30);

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, 30] (* Jean-François Alcover, May 13 2020, after Maple *)

a(n) = A306814(2n,n).

a(n) ~ c * d^n / n^2, where d = 6.7004802541941947450873... and c = 0.5171899701803656646... - Vaclav Kotesovec, Apr 13 2019

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A199915 Triangle read by rows: T(n,k) is the number 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).

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