A243752 -id:A243752 - cp's OEIS frontend

A091958 Triangle read by rows: T(n,k)=number of ordered trees with n edges and k branch nodes at odd height.

1, 1, 2, 4, 1, 9, 5, 21, 21, 51, 78, 3, 127, 274, 28, 323, 927, 180, 835, 3061, 954, 12, 2188, 9933, 4510, 165, 5798, 31824, 19734, 1430, 15511, 100972, 81684, 9790, 55, 41835, 317942, 324246, 57876, 1001, 113634, 995088, 1245762, 309036, 10920

Offset: 0

Emeric Deutsch, Mar 13 2004

T(3n,n) = binomial(3n,n)/(2n+1) = A001764(n); T(n,0) = A001006(n) (the Motzkin numbers); T(n,1) = A055219(n-3) (n>=3; most probably); Row sums are the Catalan numbers (A000108).
T(n,k) = number of ordered trees on n edges with k vertices of outdegree at least 3; T(n,k) = number of ordered trees on n edges with k vertices V such that V's rightmost descendant leaf is at distance exactly 3 from V. - David Callan, Oct 24 2004
T(n,k) is the number of Dyck n-paths containing k UUUDs. For example, T(6,2) = 3 because UUUDUUUDDDDD, UUUDDUUUDDDD, UUUDDDUUUDDD each contains 2 UUUDs. - David Callan, Nov 04 2004

			T(3,1) = 1 because the only tree having 3 edges and 1 branch node at an odd level is the tree having the shape of the letter Y.
Triangle begins:
1;
1;
2;
4,       1;
9,       5;
21,     21;
51,     78,    3;
127,   274,   28;
323,   927,  180;
835,  3061,  954,  12;
2188, 9933, 4510, 165;

Alois P. Heinz, Rows n = 0..250, flattened
E. Deutsch, A bijection on ordered trees and its consequences, J. Comb. Theory, A, 90, 210-215, 2000.
A. Kuznetsov, I. Pak, A. Postnikov, Trees associated with the Motzkin numbers, J. Comb. Theory, A, 76, 145-147, 1996.
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Cf. A001764, A001006, A055219, A243752.

Topmost entries in each column form A001764=( binomial(3n, n)/(2n+1) )A025174=(%20binomial(3n+2,%20n)%20)">(n>=0), next to topmost entries form A025174=( binomial(3n+2, n) )(n>=0), next lower entries are given by ( (n+2)binomial(3n+4, n) )_(n>=0).

T := (n,k)->binomial((n+1),k)*sum((-1)^j*binomial(n+1-k,j)*binomial(2*n-3*k-3*j,n),j=0..floor(n/3)-k)/(n+1): seq(seq(T(n,k),k=0..floor(n/3)),n=0..18);
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, expand(b(x-1, y+1, [2, 3, 4, 4][t])
      +b(x-1, y-1, [1, 1, 1, 1][t])*`if`(t=4, z, 1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..15);  # Alois P. Heinz, Jun 10 2014

Clear[a]; a[n_, k_]/;k>n/3 || k<0 := 0; a[n_, 0]/;0<=n<=1 := 1; a[n_, 0]/;n>=2 := a[n, 0] = ((2*n + 1)*a[n-1, 0] + 3*(n - 1)*a[n-2, 0])/(n + 2); a[n_, k_]/;1<=k<=n/3 && n>=2 := a[n, k] = ( (12 - 9*k + 3*n)*a[n-2, k-2] - (12 - 18*k + 3*n)*a[ n-2, k-1] - 9*k*a[ n-2, k] + (4 - 6*k + 4*n)*a[n-1, k-1] + 6*k*a[n-1, k] - (2 - k + n)*a[n, k-1] )/k; Table[a[n, k], {n, 0, 16}, {k, 0, n/3}] (Callan)
T[n_, k_] := (2*n-3*k)!*HypergeometricPFQ[{k-n-1, k-n/3, 1/3+k-n/3, 2/3+k-n/3}, {k-2*n/3, 1/3+k-2*n/3, 2/3+k-2*n/3}, 1]/(k!*(n-k+1)!*(n-3*k)!); Table[T[n, k], {n, 0, 15}, {k, 0, n/3}] // Flatten (* Jean-François Alcover, Mar 31 2015 *)

T(n,k) = binomial((n+1), k)*sum((-1)^j*binomial(n+1-k,j)*binomial(2n-3k-3j, n), j=0..floor(n/3)-k)/(n+1). G.f.: G=G(t,z) satisfies (t-1)z^3 G^3 + zG^2 - G + 1 = 0.

A092107 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UUU's (triple rises) where U=(1,1). Rows have 1,1,1,2,3,4,5,... entries, respectively.

Original entry on oeis.org

1, 1, 2, 4, 1, 9, 4, 1, 21, 15, 5, 1, 51, 50, 24, 6, 1, 127, 161, 98, 35, 7, 1, 323, 504, 378, 168, 48, 8, 1, 835, 1554, 1386, 750, 264, 63, 9, 1, 2188, 4740, 4920, 3132, 1335, 390, 80, 10, 1, 5798, 14355, 17028, 12507, 6237, 2200, 550, 99, 11, 1, 15511, 43252, 57816

Offset: 0

Emeric Deutsch, Mar 29 2004

Column 0 gives the Motzkin numbers (A001006), column 1 gives A014532. Row sums are the Catalan numbers (A000108).

Equal to A171380*B (without the zeros), B = A007318. - Philippe Deléham, Dec 10 2009

			T(5,2) = 5 because we have (U[UU)U]DUDDDD, (U[UU)U]DDUDDD, (U[UU)U]DDDUDD, (U[UU)U]DDDDUD and UD(U[UU)U]DDDD, where U=(1,1), D=(1,-1); the triple rises are shown between parentheses.
[1],[1],[2],[4, 1],[9, 4, 1],[21, 15, 5, 1],[51, 50, 24, 6, 1],[127, 161, 98, 35, 7, 1]
Triangle starts:
     1;
     1;
     2;
     4,    1;
     9,    4,    1;
    21,   15,    5,    1;
    51,   50,   24,    6,    1;
   127,  161,   98,   35,    7,    1;
   323,  504,  378,  168,   48,    8,    1;
   835, 1554, 1386,  750,  264,   63,    9,    1;
  2188, 4740, 4920, 3132, 1335,  390,   80,   10,    1;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 14.
Michael Bukata, Ryan Kulwicki, Nicholas Lewandowski, Lara Pudwell, Jacob Roth, and Teresa Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv:1812.07112 [math.CO], 2018.
FindStat - Combinatorial Statistic Finder, The number of occurrences of the contiguous pattern [.,[.,[.,.]]].
Lara Pudwell, On the distribution of peaks (and other statistics), 16th International Conference on Permutation Patterns, Dartmouth College, 2018.
Toufik Mansour and Mark Shattuck, Counting occurrences of subword patterns in non-crossing partitions, Art Disc. Appl. Math. (2022).
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, and Kostas Manes, Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property, Electr. J. Combinatorics, 19 (2012), #P2.
Yidong Sun, The statistic "number of udu's" in Dyck paths, Discrete Math., 287 (2004), 177-186.

Cf. A000108, A001006, A001405, A007318, A014532, A033321, A171380, A243752, A243753.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, expand(b(x-1, y-1, min(t+1,2))*
      `if`(t=2, z, 1) +b(x-1, y+1, 0))))
    end:
T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 0)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 11 2014

b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, Expand[b[x-1, y-1, Min[t+1, 2]]*If[t == 2, z, 1] + b[x-1, y+1, 0]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Alois P. Heinz *)

G.f.: G(t, z) satisfies z(t+z-tz)G^2 - (1-z+tz)G + 1 = 0.

Sum_{k=0..n} T(n,k)*x^k = A001405(n), A001006(n), A000108(n), A033321(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Dec 10 2009

A098978 Triangle read by rows: T(n,k) is number of Dyck n-paths with k UUDDs, 0 <= k <= n/2.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 8, 1, 13, 23, 6, 35, 69, 27, 1, 97, 212, 110, 10, 275, 662, 426, 66, 1, 794, 2091, 1602, 360, 15, 2327, 6661, 5912, 1760, 135, 1, 6905, 21359, 21534, 8022, 945, 21, 20705, 68850, 77685, 34840, 5685, 246, 1, 62642, 222892, 278192, 146092

Offset: 0

David Callan, Oct 24 2004

T(n,k) is the number of Łukasiewicz paths of length n having k peaks. A Łukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1). Example: T(3,1)=3 because we have HUD, UDH and U(2)DD, where H=(1,0), U(1,1), U(2)=(1,2) and D=(1,-1). (see R. P. Stanley reference). - Emeric Deutsch, Jan 06 2005

			Table begins
\ k  0,   1,   2, ...
n
0 |  1;
1 |  1;
2 |  1,   1;
3 |  2,   3;
4 |  5,   8,   1;
5 | 13,  23,   6;
6 | 35,  69,  27,  1;
7 | 97, 212, 110, 10;
8 |275, 662, 426, 66, 1;
T(3,1) = 3 because each of UUUDDD, UDUUDD, UUDDUD has one UUDD.

R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps. - Emeric Deutsch, Jan 06 2005

Alois P. Heinz, Rows n = 0..200, flattened
Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924. - From _N. J. A. Sloane_, May 05 2012
Index entries for sequences related to Łukasiewicz

Column k=0 is A025242 (apart from first term).

Cf. A243752.

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, expand(b(x-1, y+1, [2, 3, 3, 2][t])
      +b(x-1, y-1, [1, 1, 4, 1][t])*`if`(t=4, z, 1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..15);  # Alois P. Heinz, Jun 10 2014

T[n_, k_] := Binomial[n-k, k] Binomial[2n-3k, n-k-1] HypergeometricPFQ[{k -n/2-1/2, k-n/2, k-n/2, k-n/2+1/2}, {k-2n/3, k-2n/3+1/3, k-2n/3+2/3}, 16/27]/(n-k); T[0, 0] = 1; Flatten[Table[T[n, k], {n, 0, 15}, {k, 0, n/2}]] (* Jean-François Alcover, Dec 21 2016, after 2nd formula *)

G.f.: (1 + z^2 - t*z^2 - (-4*z + (-1 - z^2 + t*z^2)^2)^(1/2))/(2*z) = Sum_{n>=0, 0<=k<=n/2} T(n, k)z^n*t^k and it satisfies G = 1 + G^2*z + G*(-z^2 + t*z^2).

T(n,k) = Sum_{j=0..floor(n/2)-k} (-1)^j * binomial(n-(j+k), j+k) * binomial(2n-3(j+k), n-(j+k)-1) * binomial(j+k, k)/(n-(j+k)). - I. Tasoulas (jtas(AT)unipi.gr), Feb 19 2006

A114848 Triangle read by rows T(n,k) = the number of Dyck paths of semilength n with k UUDDU's, 0<=k<=[(n-1)/2].

Original entry on oeis.org

1, 1, 2, 4, 1, 10, 4, 28, 13, 1, 82, 44, 6, 248, 153, 27, 1, 770, 536, 116, 8, 2440, 1889, 486, 46, 1, 7858, 6696, 1992, 240, 10, 25644, 23849, 8042, 1180, 70, 1, 84618, 85276, 32124, 5552, 430, 12, 281844, 305933, 127287, 25306, 2430, 99, 1, 946338, 1100692

Offset: 0

I. Tasoulas (jtas(AT)unipi.gr), Feb 20 2006

Row sums are Catalan numbers A000108.

			T(4,1) = 4 because there exist 4 Dyck paths with one occurrence of UUDDU : UDUUDDUD, UUDDUDUD, UUDDUUDD, UUUDDUDD.
Triangle begins:
:  0 :     1;
:  1 :     1;
:  2 :     2;
:  3 :     4,     1;
:  4 :    10,     4;
:  5 :    28,    13,     1;
:  6 :    82,    44,     6;
:  7 :   248,   153,    27,    1;
:  8 :   770,   536,   116,    8;
:  9 :  2440,  1889,   486,   46,   1;
: 10 :  7858,  6696,  1992,  240,  10;
: 11 : 25644, 23849,  8042, 1180,  70,  1;
: 12 : 84618, 85276, 32124, 5552, 430, 12;

Alois P. Heinz, Rows n = 0..200, flattened
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924. - From _N. J. A. Sloane_, May 05 2012

Cf. A187256, A243752.

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, expand(b(x-1, y+1, [2, 3, 3, 2, 2][t])
      *`if`(t=5, z, 1) +b(x-1, y-1, [1, 1, 4, 5, 1][t]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..15);  # Alois P. Heinz, Jun 10 2014

For[n = 1, n <= 20, n++, For[k = 0, k <= Floor[(n - 1)/2], k++, Print[Sum[(-1)^j * Binomial[n - 1 - (j + k), j + k] * Binomial[j + k, k] * Binomial[2(n - 2(j + k)), n - 2(j + k)]/(n - 2(j + k) + 1), {j, 0, Floor[(n - 1)/2] - k}]]]]

T(n,k) = Sum((-1)^j * binomial(n-1-(j+k), j+k) * binomial(j + k, k) * A000108(n-2(j+k)), j=0..[(n-1)/2]-k).

G.f. G = G(t,z) satisfies G = C(z/(z^2(1-t)+1)), where C(z) is g.f. of Catalan numbers.

A116424 Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k UDUU's, 0 <= k <= floor((n-1)/2).

Original entry on oeis.org

1, 1, 2, 4, 1, 9, 5, 22, 19, 1, 57, 66, 9, 154, 221, 53, 1, 429, 729, 258, 14, 1223, 2391, 1131, 116, 1, 3550, 7829, 4652, 745, 20, 10455, 25638, 18357, 4115, 220, 1, 31160, 84033, 70404, 20598, 1790, 27, 93802, 275765, 264563, 96286, 12104, 379, 1, 284789

Offset: 0

I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006

T(n,k) also gives the number of Dyck paths of semilength n with k UUDU's.

Column k=0 gives A105633(n-1) for n > 0.

			Triangle begins:
00 :     1;
01 :     1;
02 :     2;
03 :     4,    1;
04 :     9,    5;
05 :    22,   19,    1;
06 :    57,   66,    9;
07 :   154,  221,   53,   1;
08 :   429,  729,  258,  14;
09 :  1223, 2391, 1131, 116,  1;
10 :  3550, 7829, 4652, 745, 20;
...
T(4,1) = 5 because there exist five Dyck paths of semilength 4 with one occurrence of UDUU : UDUUUDDD, UDUUDUDD, UDUUDDUD, UUDUUDDD, UDUDUUDD.

Alois P. Heinz, Rows n = 0..200, flattened
Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See p. 18.
Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Cf. A105633, A243752.

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 2][t])*
     `if`(t=4, z, 1) +b(x-1, y-1, [1, 3, 1, 3][t]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..15);  # Alois P. Heinz, Jun 02 2014

s = Series[((1 + (t - 1) z^2) - Sqrt[(1 + (t - 1) z^2)^2 - 4*z*(1 - z + z*t)])/(2*z*(1 - z + z*t)), {z, 0, 15}] // CoefficientList[#, z]&;
CoefficientList[#, t]& /@ s // Flatten (* updated by Jean-François Alcover, Feb 14 2021 *)

T(n,k) = Sum_{i=k..floor((n-1)/2)} (-1)^(i+k) * binomial(i,k) * binomial(n-i,i) * binomial(2*n-3*i, n - 2*i -1)/(n-i), n >= 1.

G.f. G = G(t,z) satisfies G = 1 + z^2(1-t)G + z(1-z+tz)G^2.

A135306 Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k UDDU's.

Original entry on oeis.org

1, 1, 2, 4, 1, 9, 5, 23, 17, 2, 63, 54, 15, 178, 177, 69, 5, 514, 594, 273, 49, 1515, 1997, 1056, 280, 14, 4545, 6698, 4077, 1308, 168, 13827, 22487, 15545, 5745, 1140, 42, 42540, 75701, 58377, 24695, 6105, 594, 132124, 255455, 216864, 103862, 29810

Offset: 0

N. J. A. Sloane, Dec 07 2007

From Emeric Deutsch, Dec 15 2007: (Start)
Row 0 has 1 term; row n (n >= 1) has ceiling(n/2) terms.
Row sums yield the Catalan numbers (A000108).
Column 0 yields A135307.
T(2n+1, n) = binomial(2n,n)/(n+1) (the Catalan numbers, A000108). (End)

			Triangle begins:
     1;
     1;
     2;
     4,    1;
     9,    5;
    23,   17,    2;
    63,   54,   15;
   178,  177,   69,    5;
   514,  594,  273,   49;
  1515, 1997, 1056,  280,   14;
  4545, 6698, 4077, 1308,  168;
  ...
T(4,1) = 5 because we have U(UDDU)DUD, U(UDDU)UDD, UU(UDDU)DD, UDU(UDDU)D and UUD(UDDU)D (the UDDU's are shown between parentheses).

Alois P. Heinz, Rows n = 0..200, flattened
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Cf. A000108, A135307, A243752.

A135306 := proc(n,k) if n =0 then 1 ; else add((-1)^(j-k)*binomial(n-k,j-k)*binomial(2*n-3*j,n-j+1),j=k..floor((n-1)/2)) ; %*binomial(n,k)/n ; fi ; end: for n from 0 to 20 do for k from 0 to max(0,(n-1)/2) do printf("%a, ",A135306(n,k)) ; od: od: # R. J. Mathar, Dec 08 2007
T:=proc(n,k) options operator, arrow: binomial(n,k)*(sum((-1)^(j-k)*binomial(n-k,j-k)*binomial(2*n-3*j,n-j+1),j=k..floor((1/2)*n-1/2)))/n end proc: 1; for n to 13 do seq(T(n,k),k=0..ceil((n-2)*1/2)) end do; # yields sequence in triangular form; Emeric Deutsch, Dec 15 2007

T[n_, k_] := Binomial[n, k]*Sum[(-1)^(j-k)*Binomial[n-k, j-k]*Binomial[2*n - 3*j, -j+n+1], {j, k, (n-1)/2}]/n; T[0, 0] = 1; Table[T[n, k], {n, 0, 13}, {k, 0, If[n == 0, 0, Quotient[n-1, 2]]}] // Flatten (* Jean-François Alcover, Nov 27 2014, after Emeric Deutsch *)

From Emeric Deutsch, Dec 15 2007: (Start)
T(n,k) = (1/n)*binomial(n,k)*Sum_{j=k..floor((n-1)/2)} (-1)^(j-k)*binomial(n-k, j-k)*binomial(2n-3j, n-j+1).
G.f.: G = G(t,z) satisfies z*G^3 - ((1-t)*z+1)*G^2 + (1+2*(1-t)*z)*G - (1-t)*z = 0. (End)

More terms from R. J. Mathar and Emeric Deutsch, Dec 08 2007

A242450 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UUDDUDUUUUDUDDDDUUDD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-2)/8)), read by rows.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16795, 1, 58783, 3, 208002, 10, 742865, 35, 2674314, 126, 9694383, 462, 35355954, 1716, 129638355, 6435, 477614391, 24308, 1, 1767170815, 92372, 3, 6563767715, 352694, 11, 24464914983, 1351996, 41, 91477363496, 5199988

Offset: 0

Alois P. Heinz, Jun 12 2014

UUDDUDUUUUDUDDDDUUDD is a Dyck path that contains all sixteen consecutive step patterns of length 4.

			Triangle T(n,k) begins:
:  0 :           1;
:  1 :           1;
:  2 :           2;
:  3 :           5;
:  4 :          14;
:  5 :          42;
:  6 :         132;
:  7 :         429;
:  8 :        1430;
:  9 :        4862;
: 10 :       16795,       1;
: 11 :       58783,       3;
: 12 :      208002,      10;
: 13 :      742865,      35;
: 14 :     2674314,     126;
: 15 :     9694383,     462;
: 16 :    35355954,    1716;
: 17 :   129638355,    6435;
: 18 :   477614391,   24308,  1;
: 19 :  1767170815,   92372,  3;
: 20 :  6563767715,  352694, 11;
: 21 : 24464914983, 1351996, 41;

Alois P. Heinz, Rows n = 0..500, flattened

Row sums give A000108.
T(834828,k) = A243752(834828,k).
T(n,0) = A243753(n,834828).
Cf. A243820.

b:= proc(x, y, t) option remember; `if`(x=0, 1,
     expand(`if`(y>=x-1, 0, b(x-1, y+1, [2, 3, 3, 2, 6, 3,
       8, 9, 10, 11, 3, 13, 3, 2, 2, 2, 18, 19, 3, 2][t]))+
     `if`(t=20, z, 1)*`if`(y=0, 0, b(x-1, y-1, [1, 1, 4, 5, 1, 7,
       1, 1, 4, 4, 12, 5, 14, 15, 16, 17, 1, 1, 20, 5][t]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..30);

b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Expand[If[y >= x - 1, 0, b[x - 1, y + 1, {2, 3, 3, 2, 6, 3, 8, 9, 10, 11, 3, 13, 3, 2, 2, 2, 18, 19, 3, 2}[[t]]]] + If[t == 20, z, 1]*If[y == 0, 0, b[x - 1, y - 1, {1, 1, 4, 5, 1, 7, 1, 1, 4, 4, 12, 5, 14, 15, 16, 17, 1, 1, 20, 5}[[t]]]]]];
T[n_] := CoefficientList[b[2n, 0, 1], z];
T /@ Range[0, 30] // Flatten (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)

A243754 Number of Dyck paths of semilength n avoiding the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1).

Original entry on oeis.org

1, 0, 0, 1, 1, 9, 1, 127, 323, 1515, 4191, 10455, 20705, 93802, 113634, 3219205, 10626023, 45980364, 139604903, 555857157, 1334821448, 7577098816, 20676558270, 61994003643, 193904367362, 800928670232, 2374027931492, 12506574770693, 29311991924792

Offset: 0

Alois P. Heinz, Jun 09 2014

nonn

			a(5) = 9 because there are 9 Dyck paths of semilength 5 avoiding the consecutive step pattern UDU given by the binary expansion of 5 = 101_2: UUDDUUDDUD, UUDDUUUDDD, UUUDDDUUDD, UUUDDUDDUD, UUUDDUUDDD, UUUUDDDDUD, UUUUDDDUDD, UUUUDDUDDD, UUUUUDDDDD.
a(6) = 1: UDUDUDUDUDUD.

Alois P. Heinz, Table of n, a(n) for n = 0..512

Column k=0 of A243752.

Main diagonal of A243753.

A243770 Number of Dyck paths of semilength n having exactly one occurrence of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1).

Original entry on oeis.org

1, 1, 3, 11, 16, 57, 161, 927, 1997, 5539, 25638, 68850, 275765, 995088, 2784600, 19235059, 53549250, 177389053, 711629836, 2641203240, 7517769634, 31706388438, 147201204924, 455738363552, 1614252170849, 6020919907344, 23811404216400, 79787485940824

Offset: 1

Alois P. Heinz, Jun 10 2014

nonn

			a(1) = 1: (U)D.
a(2) = 1: U(UD)D.
a(3) = 3: UD(UU)DD, (UU)DDUD, (UU)DUDD.
a(4) = 11: UDUDU(UDD), UDU(UDD)UD, UDUUD(UDD), UDUU(UDD)D, U(UDD)UDUD, UUD(UDD)UD, UUDUD(UDD), UUDU(UDD)D, UU(UDD)DUD, UUUD(UDD)D, UUU(UDD)DD.

Alois P. Heinz, Table of n, a(n) for n = 1..400

Column k=1 of A243752.

Main diagonal of A243827.

A243771 Number of Dyck paths of semilength n having exactly two (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).

Original entry on oeis.org

1, 1, 2, 12, 69, 98, 180, 1056, 3967, 18357, 77685, 264563, 1245762, 1915056, 5303208, 24548040, 107835695, 375494210, 1898502240, 4942470942, 23489565822, 104559681798, 413327570240, 1426320927138, 6025235528016, 19911812844324, 87316285518504

Offset: 2

Alois P. Heinz, Jun 10 2014

nonn

			a(2) = 1: (UD)[UD].
a(3) = 1: (U[U)U]DDD.
a(4) = 2: U(UDD)U[UDD], UU(UDD)[UDD].
a(5) = 12: (UD[U)DU]UDDUD, (UD[U)DU]UUDDD, (UDU)UDD[UDU]D, (UDU)[UDU]DDUD, (UDU)[UDU]UDDD, (UDU)U[UDU]DDD, UUDD(UD[U)DU]D, U(UDU)DD[UDU]D, U(UD[U)DU]DDUD, U(UD[U)DU]UDDD, U(UDU)[UDU]DDD, UU(UD[U)DU]DDD.

Alois P. Heinz, Table of n, a(n) for n = 2..350

Column k=2 of A243752.

Main diagonal of A243828.

cp's OEIS Frontend

A091958 Triangle read by rows: T(n,k)=number of ordered trees with n edges and k branch nodes at odd height.

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A092107 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UUU's (triple rises) where U=(1,1). Rows have 1,1,1,2,3,4,5,... entries, respectively.

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A098978 Triangle read by rows: T(n,k) is number of Dyck n-paths with k UUDDs, 0 <= k <= n/2.

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A114848 Triangle read by rows T(n,k) = the number of Dyck paths of semilength n with k UUDDU's, 0<=k<=[(n-1)/2].

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A116424 Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k UDUU's, 0 <= k <= floor((n-1)/2).

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A135306 Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k UDDU's.

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A242450 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UUDDUDUUUUDUDDDDUUDD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-2)/8)), read by rows.

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