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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 June 15 15:16 EDT 2021. Contains 345049 sequences. (Running on oeis4.)