A223092 Triangle read by rows: let T(n,k) (for n >= 0, 0 <= k <= n) be the number of walks from (0,0) to (n,k) using steps (1,1), (1,0), (1,-1) and (0,-1); n-th row of triangle gives T(n,n), T(n,n-1), ..., T(n,0).

1, 1, 2, 1, 4, 7, 1, 6, 18, 29, 1, 8, 33, 86, 133, 1, 10, 52, 179, 431, 650, 1, 12, 75, 316, 978, 2238, 3319, 1, 14, 102, 505, 1874, 5406, 11941, 17498, 1, 16, 133, 754, 3235, 11020, 30241, 65086, 94525, 1, 18, 168, 1071, 5193, 20202, 64698, 171045, 360897, 520508, 1, 20, 207, 1464, 7896, 34362, 124455, 380400, 977040, 2029490, 2910895

Offset

0,3

Links

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

M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013

N. J. A. Sloane, Illustration of initial terms of A223092 and A064641

Formula

T(n,k) = T(n,k+1) + T(n-1,k+1) + T(n-1,k) + T(n-1,k-1). - Philippe Deléham, Mar 29 2013

Example

Triangle begins:
[1]
[1, 2]
[1, 4, 7]
[1, 6, 18, 29]
[1, 8, 33, 86, 133]
[1, 10, 52, 179, 431, 650]
[1, 12, 75, 316, 978, 2238, 3319]
...
The T(n,k) array begins:
4:  0  0  0  0   1  10 ...
3:  0  0  0  1   8  52 ...
2:  0  0  1  6  33 179 ...
1:  0  1  4 18  86 431 ...
0:  1  2  7 29 133 650 ...
-------------------------
k/n:0  1  2  3   4   5 ...
T(5,2) = T(5,3) + T(4,3) + T(4,2) + T(4,1) = 52 + 8 + 33 + 86 = 179.- Philippe Deléham, Mar 29 2013
This is also Dziemianczuk's array N(-i,i+j) read by antidiagonals:
1,2,7,29,133,650,3319,17498, ...
1,4,18,86,431,2238,11941,65086, ...
1,6,33,179,978,5406,30241,171045, ...
1,8,52,316,1874,11020,64698,380400, ...
1,10,75,505,3235,20202,124455,761160, ...
... - N. J. A. Sloane, Dec 05 2013

Maple

T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
      `if`(n<0 or k<0 or k>n, 0, add(T(n-l[1], k-l[2]),
       l=[[1, 1], [1, 0], [1, -1], [0, -1]]) ))
    end:
seq(seq(T(n, n-j), j=0..n), n=0..10);  # Alois P. Heinz, Apr 08 2013

Mathematica

max = 10; T[0, 0] = 1; T[n_ /; n >= 0, k_ /; 0 <= k <= max] := T[n, k] = T[n, k+1]+T[n-1, k+1]+T[n-1, k]+T[n-1, k-1]; T[n_, k_] = 0; Table[Table[T[n, k], {k, n, 0, -1}], {n, 0, max}] // Flatten (* Jean-François Alcover, Mar 07 2014, after Philippe Deléham *)

Crossrefs

Cf. A064641 (T(n,0)), A071943, A052709.

Sequence in context: A341696 A052566 A234946 * A071948 A193589 A187115

Adjacent sequences: A223089 A223090 A223091 * A223093 A223094 A223095

Keyword

nonn,tabl

Author

N. J. A. Sloane, Mar 29 2013

Status

approved