A243833 -id:A243833 - cp's OEIS frontend

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.

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

Alois P. Heinz, Jun 09 2014

			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]

Alois P. Heinz, Rows n = 0..270, flattened

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.
Row sums give A000108.
Cf. A098978, A114463, A114848, A116424, A135305, A242450, A243366, A243838.

A243753 Number A(n,k) of Dyck paths of semilength n avoiding 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

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 2, 1, 4, 1, 1, 0, 0, 0, 1, 1, 2, 4, 1, 9, 1, 1, 0, 0, 0, 1, 1, 2, 4, 9, 1, 21, 1, 1, 0, 0, 0, 1, 1, 1, 4, 9, 21, 1, 51, 1, 1, 0, 0, 0

Offset: 0

Alois P. Heinz, Jun 09 2014

			Square array A(n,k) begins:
  1, 1, 1, 1, 1,   1, 1,   1,   1,    1, ...
  0, 0, 0, 1, 1,   1, 1,   1,   1,    1, ...
  0, 0, 0, 1, 1,   1, 1,   2,   2,    2, ...
  0, 0, 0, 1, 1,   2, 1,   4,   4,    4, ...
  0, 0, 0, 1, 1,   4, 1,   9,   9,    9, ...
  0, 0, 0, 1, 1,   9, 1,  21,  21,   23, ...
  0, 0, 0, 1, 1,  21, 1,  51,  51,   63, ...
  0, 0, 0, 1, 1,  51, 1, 127, 127,  178, ...
  0, 0, 0, 1, 1, 127, 1, 323, 323,  514, ...
  0, 0, 0, 1, 1, 323, 1, 835, 835, 1515, ...

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

Columns give: 0, 1, 2: A000007, 3, 4, 6: A000012, 5: A001006(n-1) for n>0, 7, 8, 14: A001006, 9: A135307, 10: A078481 for n>0, 11, 13: A105633(n-1) for n>0, 12: A082582, 15, 16: A036765, 19, 27: A114465, 20, 24, 26: A157003, 21: A247333, 25: A187256(n-1) for n>0.
Main diagonal gives A243754 or column k=0 of A243752.
Cf. A242450, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.

A:= proc(n, k) option remember; local b, m, r, h;
      if k<2 then return `if`(n=0, 1, 0) fi;
      m:= iquo(k, 2, 'r'); h:= 2^ilog2(k); b:=
      proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
        `if`(t=m and r=1, 0, b(x-1, y+1, irem(2*t+1, h)))+
        `if`(t=m and r=0, 0, b(x-1, y-1, irem(2*t, h)))))
      end; forget(b);
      b(2*n, 0, 0)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_] := A[n, k] = Module[{b, m, r, h}, If[k<2, Return[If[n == 0, 1, 0]]]; {m, r} = QuotientRemainder[k, 2]; h = 2^Floor[Log[2, k]]; b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, If[t == m && r == 1, 0, b[x-1, y+1, Mod[2*t+1, h]]] + If[t == m && r == 0, 0, b[x-1, y-1, Mod[2*t, h]]]]]; b[2*n, 0, 0]]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)

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.

Original entry on oeis.org

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

Alois P. Heinz, Jun 11 2014

			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, ...

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

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.

A243828 Number A(n,k) of Dyck paths of semilength n having exactly two (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, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 6, 10, 0, 0, 0, 0, 0, 1, 2, 20, 15, 0, 0, 0, 0, 0, 0, 3, 15, 50, 21, 0, 0, 0, 0, 0, 0, 2, 12, 69, 105, 28, 0, 0, 0, 0, 0, 0, 1, 15, 40, 252, 196, 36, 0, 0, 0, 0, 0, 0, 0, 5, 69, 135, 804, 336, 45, 0, 0

Offset: 0

Alois P. Heinz, Jun 11 2014

			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, ...
  2, 2,  1,   0,    0,    0,    0,    0,   0,    0, ...
  0, 0,  3,   1,    0,    1,    0,    0,   0,    0, ...
  0, 0,  6,   6,    2,    3,    2,    1,   0,    0, ...
  0, 0, 10,  20,   15,   12,   15,    5,   0,    2, ...
  0, 0, 15,  50,   69,   40,   69,   24,   3,   15, ...
  0, 0, 21, 105,  252,  135,  252,   98,  28,   69, ...
  0, 0, 28, 196,  804,  441,  804,  378, 180,  273, ...
  0, 0, 36, 336, 2349, 1428, 2349, 1386, 954, 1056, ...

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

Main diagonal gives A243771 or column k=2 of A243752.

Cf. A243753, A243827, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.

A243829 Number A(n,k) of Dyck paths of semilength n having exactly three (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, 5, 0, 0, 0, 5, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 1, 20, 0, 0, 0, 0, 0, 0, 0, 10, 50, 0, 0, 0, 0, 0, 0, 1, 0, 50, 105, 0, 0, 0, 0, 0, 0, 0, 4, 5, 175, 196, 0, 0, 0, 0, 0, 0, 0, 0, 20, 56, 490, 336, 0, 0, 0, 0, 0, 0, 0, 1, 5, 80, 364, 1176, 540, 0, 0

Offset: 0

Alois P. Heinz, Jun 11 2014

			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, ...
  5, 5,   1,    0,    0,    0,    0,   0,  0,   0, ...
  0, 0,   6,    1,    0,    1,    0,   0,  0,   0, ...
  0, 0,  20,   10,    0,    4,    0,   1,  0,   0, ...
  0, 0,  50,   50,    5,   20,    5,   6,  0,   0, ...
  0, 0, 105,  175,   56,   80,   56,  35,  0,   5, ...
  0, 0, 196,  490,  364,  315,  364, 168,  0,  49, ...
  0, 0, 336, 1176, 1800, 1176, 1800, 750, 12, 280, ...

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

Main diagonal gives A243772 or column k=3 of A243752.

Cf. A243753, A243827, A243828, 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

Alois P. Heinz, Jun 11 2014

			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, ...

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

Main diagonal gives A243773 or column k=4 of A243752.

Cf. A243753, A243827, A243828, A243829, A243831, A243832, A243833, A243834, A243835, A243836.

A243831 Number A(n,k) of Dyck paths of semilength n having exactly five (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, 0, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 1, 105, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 490, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 196, 1764, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 1176, 5292, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42, 0, 5292, 13860, 0, 0

Offset: 0

Alois P. Heinz, Jun 11 2014

			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,   0, 0, 0,   0, ...
   0,  0,    0,    0,  0,    0,  0,   0, 0, 0,   0, ...
  42, 42,    1,    0,  0,    0,  0,   0, 0, 0,   0, ...
   0,  0,   15,    1,  0,    1,  0,   0, 0, 0,   1, ...
   0,  0,  105,   21,  0,    6,  0,   1, 0, 0,   1, ...
   0,  0,  490,  196,  0,   42,  0,   8, 0, 0,  13, ...
   0,  0, 1764, 1176,  0,  224,  0,  63, 0, 0,  52, ...
   0,  0, 5292, 5292, 42, 1134, 42, 390, 0, 0, 244, ...

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

Main diagonal gives A243774 or column k=5 of A243752.

Cf. A243753, A243827, A243828, A243829, A243830, A243832, A243833, A243834, A243835, A243836.

A243832 Number A(n,k) of Dyck paths of semilength n having exactly six (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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 132, 0, 0, 0, 0, 0, 0, 132, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 196, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 28, 1176, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 336, 5292, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 2520, 19404, 0, 0

Offset: 0

Alois P. Heinz, Jun 11 2014

			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,  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, ...
  132, 132,    1,    0, 0,   0, 0,  0, 0, 0,  0, ...
    0,   0,   21,    1, 0,   1, 0,  0, 0, 0,  1, ...
    0,   0,  196,   28, 0,   7, 0,  1, 0, 0,  1, ...
    0,   0, 1176,  336, 0,  56, 0,  9, 0, 0, 15, ...
    0,   0, 5292, 2520, 0, 336, 0, 80, 0, 0, 64, ...

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

Main diagonal gives A243775 or column k=6 of A243752.

Cf. A243753, A243827, A243828, A243829, A243830, A243831, A243833, A243834, A243835, A243836.

A243834 Number A(n,k) of Dyck paths of semilength n having exactly eight (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, 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, 1430, 0, 0, 0, 0, 0, 0, 0, 0, 1430, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 540, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 4950, 0, 0

Offset: 0

Alois P. Heinz, Jun 11 2014

			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, 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, 0, 0,  0, 0, ...
     0,    0,    0,   0, 0,  0, 0,  0, 0, 0,  0, 0, ...
  1430, 1430,    1,   0, 0,  0, 0,  0, 0, 0,  0, 0, ...
     0,    0,   36,   1, 0,  1, 0,  0, 0, 0,  1, 0, ...
     0,    0,  540,  45, 0,  9, 0,  1, 0, 0,  1, 0, ...
     0,    0, 4950, 825, 0, 90, 0, 11, 0, 0, 19, 0, ...

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

Main diagonal gives A243777 or column k=8 of A243752.

Cf. A243753, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243835, A243836.

A243835 Number A(n,k) of Dyck paths of semilength n having exactly nine (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, 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, 4862, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4862, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 825, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 9075, 0, 0

Offset: 0

Alois P. Heinz, Jun 11 2014

			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, 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, 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, ...
  4862, 4862,   1,  0, 0,  0, 0, 0, 0, 0, 0, 0, ...
     0,    0,  45,  1, 0,  1, 0, 0, 0, 0, 1, 0, ...
     0,    0, 825, 55, 0, 10, 0, 1, 0, 0, 1, 0, ...

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

Main diagonal gives A243778 or column k=9 of A243752.

Cf. A243753, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243836.

cp's OEIS Frontend

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.

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A243753 Number A(n,k) of Dyck paths of semilength n avoiding 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.

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Mathematica

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.

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A243828 Number A(n,k) of Dyck paths of semilength n having exactly two (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.

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A243829 Number A(n,k) of Dyck paths of semilength n having exactly three (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.

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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.

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A243831 Number A(n,k) of Dyck paths of semilength n having exactly five (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.

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A243832 Number A(n,k) of Dyck paths of semilength n having exactly six (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.

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A243834 Number A(n,k) of Dyck paths of semilength n having exactly eight (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.

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A243835 Number A(n,k) of Dyck paths of semilength n having exactly nine (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.

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