cp's OEIS Frontend

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.

%I A289352 #13 Jul 19 2017 21:38:19
%S A289352 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,
%T A289352 560,420,224,70,1176,1470,1260,840,420,126,5292,5880,4410,2520,1050,
%U A289352 252,13860,16632,13860,9240,4950,1980,462,60984,66528,49896,29568,13860,4752,924
%N 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.
%F A289352 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).
%F A289352 Row sums of T(n,k) = A005558(a(n-1)).
%F A289352 T(n,1) = A001263(T(n-1,floor(n/2)).
%F A289352 T(n,floor((n+2)/2)) = A001405(a(n-1)).
%e A289352 n\k  1    2    3    4    5
%e A289352 1    1
%e A289352 2    0    1
%e A289352 3    1    2
%e A289352 4    1    2    3
%e A289352 5    6    8    6
%e A289352 6   10   15   15   10
%e A289352 7   50   60   45   20
%e A289352 8  105  140  126   84   35
%e A289352 9  490  560  420  224   70
%e A289352 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_.
%Y A289352 Cf. A001263, A001405, A005558.
%K A289352 nonn,tabf
%O A289352 1,5
%A A289352 _Roger Ford_, Jul 03 2017