A300953 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area above the path and the area below the path, measured within the smallest enclosing rectangle based on the x-axis; triangle T(n,k), n>=0, -floor((n-1)^2/4) <= k <= floor((n-1)^2/4), read by rows.

1, 1, 2, 1, 2, 2, 2, 0, 7, 0, 5, 1, 2, 3, 6, 7, 8, 6, 6, 3, 2, 0, 9, 0, 20, 0, 35, 0, 34, 0, 25, 0, 7, 1, 2, 4, 8, 10, 17, 23, 30, 38, 43, 46, 48, 42, 41, 26, 26, 12, 8, 4, 2, 0, 11, 0, 29, 0, 63, 0, 115, 0, 176, 0, 238, 0, 255, 0, 230, 0, 169, 0, 92, 0, 41, 0, 9

Offset: 0

Author

Alois P. Heinz, Mar 16 2018

Keywords

• nonn
• tabf

Examples

			              .______.
              | /\/\ |  ,  rectangle area: 12, above path area: 5,
T(3,-1) = 1:  |/____\|  ,  below path area: 7, difference: (5-7) = 2 * (-1).
.
                 /\
                /  \
T(3,0) = 2:    /    \   /\/\/\  .
.
                /\         /\
T(3,1) = 2:    /  \/\   /\/  \  .
.
Triangle T(n,k) begins:
:                                   1                                    ;
:                                   1                                    ;
:                                   2                                    ;
:                               1,  2,  2                                ;
:                           2,  0,  7,  0,  5                            ;
:                   1,  2,  3,  6,  7,  8,  6,  6,  3                    ;
:           2,  0,  9,  0, 20,  0, 35,  0, 34,  0, 25,  0,  7            ;
:  1, 2, 4, 8, 10, 17, 23, 30, 38, 43, 46, 48, 42, 41, 26, 26, 12, 8, 4  ;
		

Links

• Alois P. Heinz, Rows n = 0..50, flattened
• Wikipedia, Counting lattice paths

Crossrefs

Row sums give A000108.

Column k=0 gives A300952.

Cf. A002620, A026741, A040001, A129182, A239927, A300322, A300996.

Formula

Sum_{k = -floor((n-1)^2/4)..floor((n-1)^2/4)} k * T(n,k) = A300996(n).

T(n,-floor((n-1)^2/4)) = A040001(n).

T(n, floor((n-1)^2/4)) = A026741(n+1) for n > 2.

T(n,k) = 0 iff n is even and k is odd or abs(k) > floor(n*(n-1)/6).