A375085 Triangle read by rows: T(n,k) is the number of ballotlike paths ending at (n, k), with 0 <= k <= n.

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

Author

Stefano Spezia, Jul 29 2024

Keywords

• nonn
• tabl

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.

Examples

			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;
  ...
		

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.

Crossrefs

Cf. A000108, A026013, A057427 (diagonal), A071724, A375086 (row sums).

Programs

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

Python

from math import isqrt
from sympy import binomial
def A375085(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    b = n-binomial(a+1,2)
    return int(binomial(c:=a-1<<1,d:=a-b-1)-binomial(c,d-1)+binomial(a-2,d+1)) if n else 0 # Chai Wah Wu, Nov 14 2024

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.