A243827 Number A(n,k) of Dyck paths of semilength n having exactly one occurrence of the consecutive step pattern given by the binary expansion of k, where 1=U=(1,1) and 0=D=(1,-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 4, 6, 1, 0, 0, 0, 0, 1, 2, 11, 10, 1, 0, 0, 0, 0, 0, 4, 6, 26, 15, 1, 0, 0, 0, 0, 0, 1, 11, 16, 57, 21, 1, 0, 0, 0, 0, 0, 1, 4, 26, 45, 120, 28, 1, 0, 0, 0, 0, 1, 1, 5, 15, 57, 126, 247, 36, 1, 0, 0

Offset

0,25

Links

Alois P. Heinz, antidiagonals n = 0..140, flattened

Example

Square array A(n,k) begins:
  0, 0, 0,  0,   0,    0,   0,    0,    0,    0, ...
  1, 1, 1,  0,   0,    0,   0,    0,    0,    0, ...
  0, 0, 1,  1,   1,    1,   1,    0,    0,    0, ...
  0, 0, 1,  3,   4,    2,   4,    1,    1,    1, ...
  0, 0, 1,  6,  11,    6,  11,    4,    5,    5, ...
  0, 0, 1, 10,  26,   16,  26,   15,   21,   17, ...
  0, 0, 1, 15,  57,   45,  57,   50,   78,   54, ...
  0, 0, 1, 21, 120,  126, 120,  161,  274,  177, ...
  0, 0, 1, 28, 247,  357, 247,  504,  927,  594, ...
  0, 0, 1, 36, 502, 1016, 502, 1554, 3061, 1997, ...

Crossrefs

Columns k=2-10 give: A000012(n) for n>0, A000217(n-1) for n>0, A000295(n-1) for n>0, A005717(n-1) for n>1, A000295(n-1) for n>0, A014532(n-2) for n>2, A108863, A244235, A244236.

Main diagonal gives A243770 or column k=1 of A243752.

Cf. A243753, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.

Sequence in context: A334743 A078529 A180017 * A059530 A193525 A049828

Adjacent sequences: A243824 A243825 A243826 * A243828 A243829 A243830

Keyword

nonn,tabl

Author

Alois P. Heinz, Jun 11 2014

Status

approved