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