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A375085
Triangle read by rows: T(n,k) is the number of ballotlike paths ending at (n, k), with 0 <= k <= n.
1
0, 0, 1, 1, 1, 1, 2, 3, 2, 1, 5, 9, 6, 3, 1, 14, 28, 21, 10, 4, 1, 42, 90, 76, 39, 15, 5, 1, 132, 297, 276, 159, 64, 21, 6, 1, 429, 1001, 1002, 643, 288, 97, 28, 7, 1, 1430, 3432, 3641, 2555, 1281, 475, 139, 36, 8, 1, 4862, 11934, 13261, 10004, 5536, 2300, 733, 191, 45, 9, 1
OFFSET
0,7
COMMENTS
A ballotlike path is a lattice path in the 1st quadrant starting at (0, 0) and ending at (n, k) which uses the steps U = (1, 1), D = (1, -1), u = (1, 0) (for upstairs or umber) and d = (1, 0) (for downstairs or denim), subject to the conditions that the umber horizontal steps do not occur at height zero and the denim horizontal steps do not occur before the first down step. See pp. 8-10 in Lazar and Linusson.
LINKS
Alexander Lazar and Svante Linusson, Two-Row Set-Valued Tableaux: Catalan+k Combinatorics, Proceedings of the 36th Conference on Formal Power Series and Algebraic Combinatorics (Bochum), Séminaire Lotharingien de Combinatoire 91B (2024) Article #80, 12 pp. See p. 10.
FORMULA
T(n,k) = binomial(2*n-2,n-k-1) - binomial(2*n-2,n-k-2) + binomial(n-2,n-k).
T(n,0) = A000108(n-1).
T(n,1) = A071724(n-1) for n > 0.
T(n+1,2) - T(n,2) = A026013(n-1) for n > 2.
EXAMPLE
Triangle begins:
0;
0, 1;
1, 1, 1;
2, 3, 2, 1;
5, 9, 6, 3, 1;
14, 28, 21, 10, 4, 1;
42, 90, 76, 39, 15, 5, 1;
132, 297, 276, 159, 64, 21, 6, 1;
...
MATHEMATICA
T[n_, k_]:=Binomial[2n-2, n-k-1]-Binomial[2n-2, n-k-2]+Binomial[n-2, n-k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten
CROSSREFS
Cf. A000108, A026013, A057427 (diagonal), A071724, A375086 (row sums).
Sequence in context: A238165 A081316 A226362 * A228549 A079893 A324646
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, Jul 29 2024
STATUS
approved