A232968 Array read by antidiagonals: T(n,k) = number of lattice paths from (0,0) to (n,k) using steps (1,0), (0,1), (1,1), (-1,1) and whose points lie entirely in the integer rectangle of lattice points {(i, j): 0 <= i <= n, 0 <= j <= k}.

1, 1, 1, 1, 4, 1, 1, 7, 12, 1, 1, 10, 33, 36, 1, 1, 13, 63, 143, 108, 1, 1, 16, 102, 341, 609, 324, 1, 1, 19, 150, 656, 1748, 2583, 972, 1, 1, 22, 207, 1115, 3860, 8773, 10945, 2916, 1, 1, 25, 273, 1745, 7376, 21756, 43653, 46367, 8748, 1, 1, 28, 348, 2573, 12809, 45801, 119948, 216434, 196417, 26244, 1

Offset

0,5

Links

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

M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, Volume 30, Issue 6, pp 1427-1452.

Formula

Dziemianczuk gives a g.f.

Example

Array begins:
1,1,1,1,1,1,1,1,1,...
1,4,12,36,108,324,972,2916,8748,...
1,7,33,143,609,2583,10945,46367,196417,...
1,10,63,341,1748,8773,43653,216434,1071483,...
1,13,102,656,3860,21756,119948,653612,3539052,...
1,16,150,1115,7376,45801,274243,1606727,9288000,...
1,19,207,1745,12809,86739,558967,3489601,21333553,...
...

Maple

b:= proc(x, y, m) option remember; `if`(x=0 and y=0, 1,
      `if`(x>0, b(x-1, y, m), 0)+`if`(y>0, b(x, y-1, m), 0)+
      `if`(x>0 and y>0, b(x-1, y-1, m), 0)+
      `if`(x<m and y>0, b(x+1, y-1, m), 0))
    end:
T:= (n, k)-> b(n, k, n):
seq(seq(T(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Apr 03 2014

Mathematica

b[x_, y_, m_] := b[x, y, m] = If[x == 0 && y == 0, 1, If[x>0, b[x-1, y, m], 0] + If[y>0, b[x, y-1, m], 0] + If[x>0 && y>0, b[x-1, y-1, m], 0] + If[x<m && y>0, b[x+1, y-1, m], 0]]; T[n_, k_] := b[n, k, n]; Table[Table[T[d-k, k], {k, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Apr 24 2014, after Alois P. Heinz *)

Crossrefs

Main diagonal gives A339654.

Sequence in context: A073697 A209414 A193636 * A119673 A144447 A051455

Adjacent sequences: A232965 A232966 A232967 * A232969 A232970 A232971

Keyword

nonn,tabl

Author

N. J. A. Sloane, Dec 05 2013

Extensions

More terms from Alois P. Heinz, Apr 03 2014

Status

approved