%I #16 May 30 2022 13:25:43
%S 1,1,1,1,1,1,1,2,2,1,1,3,2,3,1,1,4,5,5,4,1,1,5,9,5,9,5,1,1,6,14,14,14,
%T 14,6,1,1,7,20,28,14,28,20,7,1,1,8,27,48,42,42,48,27,8,1,1,9,35,75,90,
%U 42,90,75,35,9,1,1,10,44,110,165,132,132,165,110,44,10,1
%N T(n,k) = M0(n+1,k,f(n,k)), where M0(p,q,r) is the number of upright paths from (0,0) to (1,0) to (p,p-q) that meet the line y = x-r and do not rise above it and f(n,k) is the least t such that M0(n+1,k,f) is not 0.
%C Let V = (e(1),...,e(n)) consist of q 1's, including e(1) = 1 and p-q 0's; let V(h) = (e(1),...,e(h)) and m(h) = (#1's in V(h)) - (#0's in V(h)) for h = 1,...,n. Then M0(p,q,r) = number of V having r = max{m(h)}.
%C f(n,k) = -1 if 0 <= k <= [(n-1)/2], else f(n,k) = 2*k-n.
%H Bruce E. Sagan and Joshua P. Swanson, <a href="https://arxiv.org/abs/2205.14078">q-Stirling numbers in type B</a>, arXiv:2205.14078 [math.CO], 2022.
%e Rows:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 2, 1;
%e 1, 3, 2, 3, 1;
%e 1, 4, 5, 5, 4, 1;
%e 1, 5, 9, 5, 9, 5, 1;
%e 1, 6, 14, 14, 14, 14, 6, 1;
%e 1, 7, 20, 28, 14, 28, 20, 7, 1;
%e 1, 8, 27, 48, 42, 42, 48, 27, 8, 1;
%e ...
%e (all palindromes)
%Y Cf. A008313.
%K nonn,tabl
%O 1,8
%A _Clark Kimberling_