A243752
Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1); triangle T(n,k), n>=0, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 1, 3, 1, 1, 11, 2, 9, 16, 12, 4, 1, 1, 57, 69, 5, 127, 161, 98, 35, 7, 1, 323, 927, 180, 1515, 1997, 1056, 280, 14, 4191, 5539, 3967, 1991, 781, 244, 64, 17, 1, 1, 10455, 25638, 18357, 4115, 220, 1, 20705, 68850, 77685, 34840, 5685, 246, 1
Offset: 0
Triangle T(n,k) begins:
: n\k : 0 1 2 3 4 5 ...
+-----+----------------------------------------------------------
: 0 : 1; [row 0 of A131427]
: 1 : 0, 1; [row 1 of A131427]
: 2 : 0, 1, 1; [row 2 of A090181]
: 3 : 1, 3, 1; [row 3 of A001263]
: 4 : 1, 11, 2; [row 4 of A091156]
: 5 : 9, 16, 12, 4, 1; [row 5 of A091869]
: 6 : 1, 57, 69, 5; [row 6 of A091156]
: 7 : 127, 161, 98, 35, 7, 1; [row 7 of A092107]
: 8 : 323, 927, 180; [row 8 of A091958]
: 9 : 1515, 1997, 1056, 280, 14; [row 9 of A135306]
: 10 : 4191, 5539, 3967, 1991, 781, 244, ... [row 10 of A094507]
Columns k=0-10 give:
A243754,
A243770,
A243771,
A243772,
A243773,
A243774,
A243775,
A243776,
A243777,
A243778,
A243779, or main diagonals of
A243753,
A243827,
A243828,
A243829,
A243830,
A243831,
A243832,
A243833,
A243834,
A243835,
A243836.
A243830
Number A(n,k) of Dyck paths of semilength n having exactly four (possibly overlapping) occurrences 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.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 1, 50, 0, 0, 0, 0, 0, 0, 0, 0, 15, 175, 0, 0, 0, 0, 0, 0, 0, 1, 0, 105, 490, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 490, 1176, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 14, 1764, 2520, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 140, 210, 5292, 4950, 0, 0
Offset: 0
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
14, 14, 1, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 10, 1, 0, 1, 0, 0, 0, 0, ...
0, 0, 50, 15, 0, 5, 0, 1, 0, 0, ...
0, 0, 175, 105, 0, 30, 0, 7, 0, 0, ...
0, 0, 490, 490, 14, 140, 14, 48, 0, 0, ...
0, 0, 1176, 1764, 210, 630, 210, 264, 0, 14, ...
Showing 1-2 of 2 results.