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.

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, 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, 42, 90, 75, 35, 9, 1, 1, 10, 44, 110, 165, 132, 132, 165, 110, 44, 10, 1

Offset

1,8

Comments

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)}.

f(n,k) = -1 if 0 <= k <= [(n-1)/2], else f(n,k) = 2*k-n.

Links

Table of n, a(n) for n=1..78.

Bruce E. Sagan and Joshua P. Swanson, q-Stirling numbers in type B, arXiv:2205.14078 [math.CO], 2022.

Example

Rows:
  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, 14,  6,  1;
  1,  7, 20, 28, 14, 28, 20,  7,  1;
  1,  8, 27, 48, 42, 42, 48, 27,  8,  1;
  ...
(all palindromes)

Crossrefs

Cf. A008313.

Sequence in context: A370489 A057555 A075532 * A047130 A125778 A047110

Adjacent sequences: A050173 A050174 A050175 * A050177 A050178 A050179

Keyword

nonn,tabl

Author

Clark Kimberling

Status

approved