A289352 Irregular triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with floor((n+2)/2) up movements in odd numbered positions and k returns to the x axis.

1, 0, 1, 1, 2, 1, 2, 3, 6, 8, 6, 10, 15, 15, 10, 50, 60, 45, 20, 105, 140, 126, 84, 35, 490, 560, 420, 224, 70, 1176, 1470, 1260, 840, 420, 126, 5292, 5880, 4410, 2520, 1050, 252, 13860, 16632, 13860, 9240, 4950, 1980, 462, 60984, 66528, 49896, 29568, 13860, 4752, 924

Offset: 1

Author

Roger Ford, Jul 03 2017

Keywords

• nonn
• tabf

Examples

			n\k  1    2    3    4    5
1    1
2    0    1
3    1    2
4    1    2    3
5    6    8    6
6   10   15   15   10
7   50   60   45   20
8  105  140  126   84   35
9  490  560  420  224   70
T(4,3)=3: (U = up in odd position, u = up in even position, d = down, _ = return to x axis, floor ((n+2)/2) = 3 up movements in odd position)  Ud_Ud_Uudd_, Uudd_Ud_Ud_, Ud_Uudd_Ud_.
		

Crossrefs

Cf. A001263, A001405, A005558.

Formula

T(1,1)=1, T(2,1)=0, T(2,2)=1, For n >= 3, T(n,k) = (1/floor((n-1)/2))*C(n-1,floor((n-3)/2))*C(n-1-k,floor((n-3)/2))*k (conjectured).

Row sums of T(n,k) = A005558(a(n-1)).

T(n,1) = A001263(T(n-1,floor(n/2)).

T(n,floor((n+2)/2)) = A001405(a(n-1)).