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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.

%I A375085 #20 Nov 15 2024 11:01:25
%S A375085 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 A375085 297,276,159,64,21,6,1,429,1001,1002,643,288,97,28,7,1,1430,3432,3641,
%U A375085 2555,1281,475,139,36,8,1,4862,11934,13261,10004,5536,2300,733,191,45,9,1
%N A375085 Triangle read by rows: T(n,k) is the number of ballotlike paths ending at (n, k), with 0 <= k <= n.
%C A375085 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 A375085 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 A375085 T(n,k) = binomial(2*n-2,n-k-1) - binomial(2*n-2,n-k-2) + binomial(n-2,n-k).
%F A375085 T(n,0) = A000108(n-1).
%F A375085 T(n,1) = A071724(n-1) for n > 0.
%F A375085 T(n+1,2) - T(n,2) = A026013(n-1) for n > 2.
%e A375085 Triangle begins:
%e A375085     0;
%e A375085     0,   1;
%e A375085     1,   1,   1;
%e A375085     2,   3,   2,   1;
%e A375085     5,   9,   6,   3,  1;
%e A375085    14,  28,  21,  10,  4,  1;
%e A375085    42,  90,  76,  39, 15,  5, 1;
%e A375085   132, 297, 276, 159, 64, 21, 6, 1;
%e A375085   ...
%t A375085 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 A375085 (Python)
%o A375085 from math import isqrt
%o A375085 from sympy import binomial
%o A375085 def A375085(n):
%o A375085     a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
%o A375085     b = n-binomial(a+1,2)
%o A375085     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 A375085 Cf. A000108, A026013, A057427 (diagonal), A071724, A375086 (row sums).
%K A375085 nonn,tabl
%O A375085 0,7
%A A375085 _Stefano Spezia_, Jul 29 2024