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A375763
Irregular triangle read by rows, T(n,k) is the number of North-East lattice paths from (0,0) to (n,n+2) that stay weakly above y = x, with weight = k + A000217(n).
1
1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 4, 5, 4, 4, 3, 2, 1, 1, 1, 4, 7, 10, 11, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 1, 5, 11, 18, 24, 27, 30, 29, 28, 25, 23, 19, 16, 12, 10, 7, 5, 3, 2, 1, 1, 1, 6, 16, 30, 46, 59, 71, 78, 81, 81, 78, 74, 67, 60, 52, 46, 37, 31, 24
OFFSET
0,6
COMMENTS
Here the weight of a lattice path is the area under the path and above the x-axis. T(n,k) also counts the number of integer compositions of (3*n) + (2*k) + 6 with adjacent differences in {-1,1}, first part 1, and last part 3.
LINKS
EXAMPLE
Triangle begins:
k=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
n=0: 1;
n=1: 1, 1, 1;
n=2: 1, 2, 2, 2, 1, 1;
n=3: 1, 3, 4, 5, 4, 4, 3, 2, 1, 1;
n=4: 1, 4, 7, 10, 11, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1;
...
T(1,0) = 1: (NENN).
T(2,1) = 2: (NNEENN) and (NENNEN).
T(3,2) = 4: (NENENNNE), (NENNENEN), (NNEENNEN), and (NNENEENN).
PROG
(Python) # see linked program
CROSSREFS
Cf. A000245 (empirical row sums), A000217 (row lengths).
Cf. A227543 (paths of this kind from (0,0) to (n,n), offset 1 for (0,0) to (n,n+1)).
Sequence in context: A302111 A124278 A253187 * A139755 A213366 A212648
KEYWORD
nonn,easy,tabf
AUTHOR
John Tyler Rascoe, Aug 26 2024
STATUS
approved