cp's OEIS Frontend

A050176 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.

%I A050176 #16 May 30 2022 13:25:43
%S A050176 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 A050176 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 A050176 42,90,75,35,9,1,1,10,44,110,165,132,132,165,110,44,10,1
%N A050176 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 A050176 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 A050176 f(n,k) = -1 if 0 <= k <= [(n-1)/2], else f(n,k) = 2*k-n.
%H A050176 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 A050176 Rows:
%e A050176   1;
%e A050176   1,  1;
%e A050176   1,  1,  1;
%e A050176   1,  2,  2,  1;
%e A050176   1,  3,  2,  3,  1;
%e A050176   1,  4,  5,  5,  4,  1;
%e A050176   1,  5,  9,  5,  9,  5,  1;
%e A050176   1,  6, 14, 14, 14, 14,  6,  1;
%e A050176   1,  7, 20, 28, 14, 28, 20,  7,  1;
%e A050176   1,  8, 27, 48, 42, 42, 48, 27,  8,  1;
%e A050176   ...
%e A050176 (all palindromes)
%Y A050176 Cf. A008313.
%K A050176 nonn,tabl
%O A050176 1,8
%A A050176 _Clark Kimberling_