A243753 -id:A243753 - cp's OEIS frontend

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.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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.

A243836 Number A(n,k) of Dyck paths of semilength n having exactly ten (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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16796, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16796, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1210, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 66, 15730, 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, ...
      0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  16796, 16796,  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
      0,     0, 55, 1, 0, 1, 0, 0, 0, 0, 1, 0, ...

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

Main diagonal gives A243779 or column k=10 of A243752.

Cf. A243753, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835.

A243881 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/4)), read by rows.

Original entry on oeis.org

1, 1, 2, 5, 14, 41, 1, 129, 3, 419, 10, 1395, 35, 4737, 124, 1, 16338, 454, 4, 57086, 1684, 16, 201642, 6305, 65, 718855, 23781, 263, 1, 2583149, 90209, 1077, 5, 9346594, 343809, 4419, 23, 34023934, 1315499, 18132, 105, 124519805, 5050144, 74368, 472, 1

Offset: 0

Alois P. Heinz, Jun 13 2014

UDUUUDDDUD is the only Dyck path of semilength 5 that contains all eight consecutive step patterns of length 3.

			Triangle T(n,k) begins:
:  0 :        1;
:  1 :        1;
:  2 :        2;
:  3 :        5;
:  4 :       14;
:  5 :       41,       1;
:  6 :      129,       3;
:  7 :      419,      10;
:  8 :     1395,      35;
:  9 :     4737,     124,     1;
: 10 :    16338,     454,     4;
: 11 :    57086,    1684,    16;
: 12 :   201642,    6305,    65;
: 13 :   718855,   23781,   263,   1;
: 14 :  2583149,   90209,  1077,   5;
: 15 :  9346594,  343809,  4419,  23;
: 16 : 34023934, 1315499, 18132, 105;

Alois P. Heinz, Rows n = 0..300, flattened

Columns k=0-10 give: A243870, A243871, A243872, A243873, A243874, A243875, A243876, A243877, A243878, A243879, A243880.
Row sums give A000108.
T(738,k) = A243752(738,k).
T(n,0) = A243753(n,738).
Cf. A243882.

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, 5, 6, 2, 4, 2, 10, 2][t])+`if`(t=10,
      z, 1)*b(x-1, y-1, [1, 3, 1, 3, 3, 7, 8, 9, 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..20);

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, 1, Expand[b[x-1, y+1, {2, 2, 4, 5, 6, 2, 4, 2, 10, 2}[[t]]] + If[t==10, z, 1]*b[x-1, y-1, {1, 3, 1, 3, 3, 7, 8, 9, 1, 3}[[t]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

A094507 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having k UDUD's (here U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 19, 14, 7, 1, 1, 53, 46, 22, 9, 1, 1, 153, 150, 82, 31, 11, 1, 1, 453, 495, 299, 127, 41, 13, 1, 1, 1367, 1651, 1087, 507, 181, 52, 15, 1, 1, 4191, 5539, 3967, 1991, 781, 244, 64, 17, 1, 1, 13015, 18692, 14442, 7824, 3271, 1128, 316, 77, 19

Offset: 0

Emeric Deutsch, Jun 05 2004

Column k=0 is A078481.
Column k=1 is A244236.
Row sums are the Catalan numbers (A000108).

			T(3,0) = 3 because UDUUDD, UUDDUD and UUUDDD are the only Dyck paths of semilength 3 and having no UDUD in them.
Triangle begins:
     1;
     1;
     1,    1;
     3,    1,    1;
     7,    5,    1,    1;
    19,   14,    7,    1,   1;
    53,   46,   22,    9,   1,   1;
   153,  150,   82,   31,  11,   1,  1;
   453,  495,  299,  127,  41,  13,  1,  1;
  1367, 1651, 1087,  507, 181,  52, 15,  1, 1;
  4191, 5539, 3967, 1991, 781, 244, 64, 17, 1, 1;

Alois P. Heinz, Rows n = 0..150, flattened
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Cf. A078481, A000108, A102405 (the same for DUDU), A243752, A243753, A244236.

T(2n,n) gives A304361.

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])
      +b(x-1, y-1, [1, 3, 1, 3][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 02 2014

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, {2, 2, 4, 2}[[t]]] + b[x-1, y-1, {1, 3, 1, 3}[[t]]]*If[t == 4, z, 1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1] ]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Alois P. Heinz *)

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

A243412 Number of Dyck paths of semilength n avoiding the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 112, 352, 1136, 3742, 12529, 42513, 145868, 505234, 1764157, 6203370, 21947490, 78072209, 279062937, 1001803617, 3610366030, 13057141261, 47373444827, 172381857939, 628944880851, 2300410562946, 8433110899963, 30980398420830, 114034887644860

Offset: 0

Alois P. Heinz, Jun 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=0 of A243366.

Column k=45 of A243753.

Recurrence: (n+1)*(n+2)*(817*n^7 - 24387*n^6 + 285094*n^5 - 1647261*n^4 + 4787137*n^3 - 5628540*n^2 - 1552284*n + 6122952)*a(n) = (n+1)*(1634*n^8 - 47957*n^7 + 542786*n^6 - 2900786*n^5 + 6449435*n^4 + 3292426*n^3 - 41693904*n^2 + 63681552*n - 24491808)*a(n-1) + 3*n*(2451*n^8 - 73161*n^7 + 850153*n^6 - 4796076*n^5 + 12712261*n^4 - 7403931*n^3 - 33886709*n^2 + 64848252*n - 30495792)*a(n-2) - (8170*n^9 - 256125*n^8 + 3222045*n^7 - 20734872*n^6 + 70290303*n^5 - 101053185*n^4 - 62925628*n^3 + 384515340*n^2 - 387509328*n + 86320944)*a(n-3) + 3*(4085*n^9 - 134190*n^8 + 1787518*n^7 - 12351340*n^6 + 46074358*n^5 - 78991732*n^4 - 20763151*n^3 + 311152124*n^2 - 443676900*n + 188645328)*a(n-4) - (8170*n^9 - 280635*n^8 + 3929664*n^7 - 28666521*n^6 + 113672493*n^5 - 215520840*n^4 + 17606573*n^3 + 648300408*n^2 - 951192216*n + 363243312)*a(n-5) + 2*(4085*n^9 - 146445*n^8 + 2159949*n^7 - 16771674*n^6 + 71463813*n^5 - 145058547*n^4 - 9273941*n^3 + 640553178*n^2 - 1114925472*n + 598040712)*a(n-6) + (8170*n^9 - 305145*n^8 + 4669113*n^7 - 37343346*n^6 + 161916525*n^5 - 325736907*n^4 - 55373986*n^3 + 1484026824*n^2 - 2345628420*n + 1080273456)*a(n-7) + (6536*n^9 - 253920*n^8 + 4039503*n^7 - 33528057*n^6 + 150519924*n^5 - 315037869*n^4 - 26105741*n^3 + 1400728128*n^2 - 2351058696*n + 1235710944)*a(n-8) + (n-9)*(6536*n^8 - 204900*n^7 + 2511339*n^6 - 14959584*n^5 + 41778954*n^4 - 25451829*n^3 - 129319352*n^2 + 282520572*n - 168563664)*a(n-9) + 3*(n-10)*(n-9)*(817*n^7 - 18668*n^6 + 155929*n^5 - 559001*n^4 + 589888*n^3 + 1351597*n^2 - 3752130*n + 2343528)*a(n-10). - Vaclav Kotesovec, Jun 05 2014

a(n) ~ c * d^n / n^(3/2), where d = 3.8821590268628506747194368909643384060073824... is the root of the equation d^8 - 2*d^7 - 10*d^6 + 12*d^5 - 5*d^4 - 2*d^3 - 5*d^2 - 8*d - 3 = 0, and c = 0.56162811676670317653498040062091920282038218... . - Vaclav Kotesovec, Jun 05 2014

A135307 Number of Dyck paths of semilength n that do not contain the string UDDU.

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 63, 178, 514, 1515, 4545, 13827, 42540, 132124, 413741, 1304891, 4141198, 13214815, 42375461, 136478383, 441285890, 1431925180, 4661485203, 15219836738, 49827678840, 163535624722, 537962562453, 1773437280323

Offset: 0

N. J. A. Sloane, Dec 07 2007

Top left terms of powers of the production matrix M generates sequence A102403. - Gary W. Adamson, Jan 30 2012

			a(6) = 63 since the top row of M^5 = (17, 17, 13, 10, 5, 1), sum of terms = 63.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric S. Egge and Kailee Rubin, Snow Leopard Permutations and Their Even and Odd Threads, arXiv:1508.05310 [math.CO], 2015
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Leading column of A135306.
Cf. A102403.
Column k=9 of A243753.

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: A135307 := proc(n) A135306(n,0) ; end: for n from 0 to 30 do printf("%a, ",A135307(n)) ; od: # R. J. Mathar, Dec 08 2007
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1$2, 2, 4][n+1],
      (2*n*(n-1)*(28*n^2-56*n-3)*a(n-1)
       +(140*n^4-630*n^3+1063*n^2-699*n+144)*a(n-2)
       -12*(n-3)*(14*n^3-42*n^2+16*n+21)*a(n-3)
       +23*(n-3)*(n-4)*(28*n^2-14*n-3)*a(n-4))/
       (n*(n+1)*(28*n^2-70*n+39)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 13 2014

a[n_] := Sum[(-1)^j*Binomial[n, j]*Binomial[2*n-3*j, n-j+1], {j, 0, (n-1)/2}]/n; a[0] = 1; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Nov 27 2014, after R. J. Mathar *)

G.f.: f(x) satisfies x*f(x)^3 - (x+1)*f(x)^2 + (2*x+1)*f(x) - x = 0 . - Eric Rowland, Mar 29 2013
The Sapounakis et al. reference gives an explicit formula.
From Gary W. Adamson, Jan 30 2012: (Start)
a(n) is the sum of top row terms in M^(n-1), where M = the following infinite square production matrix:
  1, 1, 0, 0, 0, 0, ...
  0, 1, 1, 0, 0, 0, ...
  1, 0, 1, 1, 0, 0, ...
  1, 1, 0, 1, 1, 0, ...
  1, 1, 1, 0, 1, 1, ... (End)
a(n) ~ sqrt(8 + 5*sqrt(2) + sqrt(2*(11 + 8*sqrt(2))/7))/4 * ((1 + sqrt(13 + 16*sqrt(2)))/2)^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 27 2015

More terms from R. J. Mathar, 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.

A247333 Number of Dyck paths of semilength n avoiding the consecutive step pattern UDUDU, where U=(1,1) and D=(1,-1).

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 92, 283, 893, 2875, 9407, 31189, 104555, 353794, 1206821, 4145350, 14326184, 49778473, 173794610, 609392578, 2145057797, 7577098816, 26850456704, 95425761829, 340047930692, 1214738997142, 4349231444405, 15604726428805, 56098211626478

Offset: 0

Alois P. Heinz, Sep 13 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
Beáta Bényi, Toufik Mansour, and José L. Ramírez, Set partitions and non-crossing partitions with l-neighbors and l-isolated elements, Australasian J. Comb. (2022) Vol. 84, No. 2, 325-340.

Cf. A001006.
Column k=0 of A246188.
Column k=21 of A243753.

a:= proc(n) option remember; `if`(n<5, [1$2, 2, 4, 11][n+1],
      ((n-2)*a(n-1) +(7*n-11)*a(n-2) +(11*n-31)*a(n-3)
      +(9*n-36)*a(n-4) +(3*n-15)*a(n-5)) / (n+1))
    end:
seq(a(n), n=0..40);

CoefficientList[Series[(1 + x + x^2 - Sqrt[-3 x^4 - 6 x^3 - 5 x^2 - 2 x + 1])/(2 x^2 + 2 x), {x, 0, 28}], x] (* or *)
{1}~Join~Table[Sum[(Binomial[k, n - k] Sum[Binomial[j, -k + 2 j - 1] Binomial[k, j], {j, 0, k}])/k, {k, 1, n}], {n, 1, 28}] (* Michael De Vlieger, Mar 03 2016, the latter after Maxima by Vladimir Kruchinin *)

a(n):=if n=0 then 1 else sum((binomial(k,n-k)*sum(binomial(j,-k+2*j-1)*binomial(k,j),j,0,k))/k,k,1,n); /* Vladimir Kruchinin, Mar 03 2016 */

Recursion: see Maple program.
a(n) ~ sqrt(42-6*sqrt(21)) * ((3+sqrt(21))/2)^n / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 16 2014
From Vladimir Kruchinin, Mar 03 2016: (Start)
G.f.: (1+x+x^2-sqrt(-3*x^4-6*x^3-5*x^2-2*x+1))/(2*x^2+2*x).
G.f.: B(x)+1, where B(x) satisfies B(x)=(x+x^2)*(1+B(x)+B(x)^2).
a(n) = Sum_{k=1..n} binomial(k,n-k)*M(k+1), a(0)=1, where M(k) are Motzkin numbers (A001006). (End)

cp's OEIS Frontend

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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A243836 Number A(n,k) of Dyck paths of semilength n having exactly ten (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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A243881 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/4)), read by rows.

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A094507 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having k UDUD's (here U=(1,1), D=(1,-1)).

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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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A243412 Number of Dyck paths of semilength n avoiding the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

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A135307 Number of Dyck paths of semilength n that do not contain the string UDDU.

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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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