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 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. 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 LINKS Table of n, a(n) for n=1..55. 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)). EXAMPLE 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. Sequence in context: A249050 A341649 A056493 * A277619 A001371 A339408 Adjacent sequences: A289349 A289350 A289351 * A289353 A289354 A289355 KEYWORD nonn,tabf AUTHOR Roger Ford, Jul 03 2017 STATUS approved

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Last modified July 14 19:43 EDT 2024. Contains 374323 sequences. (Running on oeis4.)