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A364386
Triangle T(n,k) read by rows: the number of Motzkin paths of length n that have k nodes at their peak level, 1 <= k <= n+1.
2
1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 7, 4, 1, 0, 1, 18, 15, 11, 5, 1, 0, 1, 44, 33, 26, 16, 6, 1, 0, 1, 113, 78, 59, 42, 22, 7, 1, 0, 1, 296, 197, 138, 101, 64, 29, 8, 1, 0, 1, 782, 518, 342, 240, 165, 93, 37, 9, 1, 0, 1, 2076, 1388, 892, 590, 406, 258, 130, 46, 10, 1, 0, 1
OFFSET
0,7
LINKS
FORMULA
T(n,n) = 1. (All nodes on level 0, only H steps.)
T(n,n-1) = 0.
T(n,n-2) = 1. (steps UHHH...HHHD)
EXAMPLE
Example for 9 paths of length n=4: UUDD (k=1 at level 2), UHHD (k=3 at level 1), UHDH (k=2 at level 1), UDUD (k=2 at level 1), UDHH (k=1 at level 1), HUHD (k=2 at level 1), HUDH (k=1 at level 1), HHUD (k=1 at level 1), HHHH (k=5 at level 0). So k=1 appears 4 times, k=2 3 times, k=3 once, k=4 never, k=5 once.
The triangle starts:
1
0, 1
1, 0, 1
2, 1, 0, 1
4, 3, 1, 0, 1
8, 7, 4, 1, 0, 1
18, 15, 11, 5, 1, 0, 1
44, 33, 26, 16, 6, 1, 0, 1
113, 78, 59, 42, 22, 7, 1, 0, 1
296, 197, 138, 101, 64, 29, 8, 1, 0, 1
782, 518, 342, 240, 165, 93, 37, 9, 1, 0, 1
2076, 1388, 892, 590, 406, 258, 130, 46, 10, 1, 0, 1
5538, 3747, 2401, 1522, 1005, 665, 388, 176, 56, 11, 1, 0, 1
14856, 10147, 6560, 4085, 2576, 1680, 1054, 564, 232, 67, 12, 1, 0, 1
...
CROSSREFS
Cf. A001006 (row sums), A088457 (column k=1).
Cf. A152879 (equivalent for Dyck paths).
Sequence in context: A119328 A058716 A048723 * A088455 A361390 A369326
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, Jul 21 2023
STATUS
approved