A375085 - OEIS

%I #20 Nov 15 2024 11:01:25

%S 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,

%T 297,276,159,64,21,6,1,429,1001,1002,643,288,97,28,7,1,1430,3432,3641,

%U 2555,1281,475,139,36,8,1,4862,11934,13261,10004,5536,2300,733,191,45,9,1

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

%C 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.

%H Alexander Lazar and Svante Linusson, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2024/80.html">Two-Row Set-Valued Tableaux: Catalan+k Combinatorics</a>, 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.

%F T(n,k) = binomial(2*n-2,n-k-1) - binomial(2*n-2,n-k-2) + binomial(n-2,n-k).

%F T(n,0) = A000108(n-1).

%F T(n,1) = A071724(n-1) for n > 0.

%F T(n+1,2) - T(n,2) = A026013(n-1) for n > 2.

%e Triangle begins:

%e     0;

%e     0,   1;

%e     1,   1,   1;

%e     2,   3,   2,   1;

%e     5,   9,   6,   3,  1;

%e    14,  28,  21,  10,  4,  1;

%e    42,  90,  76,  39, 15,  5, 1;

%e   132, 297, 276, 159, 64, 21, 6, 1;

%e   ...

%t 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

%o (Python)

%o from math import isqrt

%o from sympy import binomial

%o def A375085(n):

%o     a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))

%o     b = n-binomial(a+1,2)

%o     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

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

%K nonn,tabl

%O 0,7

%A _Stefano Spezia_, Jul 29 2024