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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
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
Sequence in context: A249050 A341649 A056493 * A277619 A001371 A339408
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.)