A000957 -id:A000957 - cp's OEIS frontend

A059027 Number of Dyck paths of semilength n with no peak at height 4.

1, 1, 2, 5, 13, 35, 97, 276, 805, 2404, 7343, 22916, 72980, 236857, 782275, 2625265, 8938718, 30834165, 107608097, 379454447, 1350434278, 4845475311, 17512579630, 63703732426, 233063976059, 857067469749, 3166309373615, 11745982220846

Offset: 0

N. J. A. Sloane, Feb 12 2001

nonn

			1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + 97*x^6 + ...

Peart and Woan, in press, G_4(x).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
P. Peart and W.-J. Woan, Dyck Paths With No Peaks at Height k, J. Integer Sequences, 4 (2001), #01.1.3.

G_1, G_2, G_3, G_4 give A000957, A000108, A059019, A059027 resp.

CoefficientList[Series[(2 - 3 x + x (1 - 4 x)^(1/2))/(2 - 5 x + x (1 - 4 x)^(1/2)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 05 2013 *)

x='x+O('x^66); Vec((2-3*x+x*(1-4*x)^(1/2))/(2-5*x+x*(1-4*x)^(1/2))) \\ Joerg Arndt, Oct 03 2013

G.f.: (2-3*x+x*(1-4*x)^(1/2))/(2-5*x+x*(1-4*x)^(1/2)).
a(n) = sum(k=1..n-2, sum(j=max(2*k-n+1,0)..k-1, (binomial(k,j)*((k-j)*binomial(2*n-3*k+j-3,n-1-2*k+j)))/(n-k-1)*2^j))+2^(n-1). - Vladimir Kruchinin, Oct 03 2013
a(n) ~ 4^n/(9*sqrt(Pi)*n^(3/2)) * (1+197/(24*n)). - Vaclav Kotesovec, Mar 20 2014

A105640 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k UUDD's, where U=(1,1) and D=(1,-1) (0<=k<=floor(n/2), n>=2). A hill in a Dyck path is a peak at level 1.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 1, 5, 10, 3, 14, 29, 13, 1, 39, 89, 52, 6, 111, 279, 195, 36, 1, 322, 881, 722, 185, 10, 947, 2806, 2637, 867, 80, 1, 2818, 8997, 9528, 3846, 520, 15, 8470, 28997, 34163, 16382, 2976, 155, 1, 25677, 93858, 121749, 67696, 15631, 1246, 21

Offset: 2

Emeric Deutsch, May 08 2006

Row n has 1+floor(n/2) terms. Row sums are the Fine numbers (A000957). T(n,0)=A105641(n). Sum(k*T(n,k),k=0..floor(n/2))=A116914(n).

			T(5,2)=3 because we have U(UUDD)(UUDD)D, (UUDD)U(UUDD)D and U(UUDD)D(UUDD) (the UUDD's are shown between parentheses).
Triangle starts:
  0,1;
  1,1;
  2,3,1;
  5,10,3;
  14,29,13,1;
  ...

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Cf. A000957, A105641, A116914.

G:=(1+2*z+z^2-t*z^2-sqrt(1-4*z+2*z^2-2*t*z^2+z^4-2*z^4*t+t^2*z^4))/2/z/(2+z+z^2-t*z^2)-1: Gser:=simplify(series(G,z=0,17)): for n from 2 to 14 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 2 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

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

A118974 Sum of the lengths of the first descents in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1).

Original entry on oeis.org

0, 0, 2, 4, 11, 31, 94, 298, 977, 3283, 11243, 39087, 137569, 489171, 1754596, 6340756, 23063731, 84372061, 310216081, 1145748061, 4248861631, 15814069951, 59054807821, 221197379221, 830819449003, 3128511421663, 11808294045071, 44666151392095, 169294875129839

Offset: 0

Emeric Deutsch, May 08 2006

nonn

			a(4)=11 because in the hill-free Dyck paths of semilength 4, namely uu(dd)uudd, uu(d)uuddd, uu(d)ududd, uuu(dd)udd, uuu(d)uddd and uuuu(dddd), the sum of the lengths of the first descents (shown between parentheses) is 2+1+1+2+1+4=11.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Cf. A000957, A118972.

F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=series(z^2*C*F*(1+C-z*C)/(1-z),z=0,32): seq(coeff(g,z,n),n=0..28);

CoefficientList[Series[x^2*(1-Sqrt[1-4*x])/2/x*(1-Sqrt[1-4*x])/x/(3-Sqrt[1-4*x])*(1+(1-Sqrt[1-4*x])/2/x-x*(1-Sqrt[1-4*x])/2/x)/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

my(x='x+O('x^50)); concat([0,0], Vec(x^2*(1-sqrt(1-4*x))/2/x*(1-sqrt(1-4*x))/x/(3-sqrt(1-4*x))*(1+(1-sqrt(1-4*x))/2/x-x*(1-sqrt(1-4*x))/2/x)/(1-x))) \\ G. C. Greubel, Mar 18 2017

a(n) = Sum_{k=1,..,n} k*A118972(n,k).
G.f.: z^2*C*F*(1+C-z*C)/(1-z), where F = (1-sqrt(1-4*z))/(z*(3-sqrt(1-4*z))) and C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
a(n) ~ 17*4^n/(27*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: 2*(n+1)*(17*n^2-65*n+60)*a(n) -3*(3*n-4)*(17*n^2-48*n+15)*a(n-1) +3*(17*n^3-82*n^2+121*n-60)*a(n-2) +2*(2*n-5) *(17*n^2-31*n+12) *a(n-3)=0. - R. J. Mathar, Jun 22 2016

A119012 Number of valleys strictly above the x-axis in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1).

Original entry on oeis.org

0, 0, 1, 5, 23, 98, 405, 1644, 6604, 26356, 104746, 415155, 1642493, 6490622, 25629581, 101156936, 399151400, 1574818496, 6213255614, 24515233082, 96739530062, 381803092580, 1507141137026, 5950525214360, 23498966702808

Offset: 1

Emeric Deutsch, May 08 2006

nonn

			a(4)=5 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UUUUDDDD, UUUD|UDDD, UUD|UUDDD, UUD|UD|UDD, UUUDD|UDD and UUDDUUDD (U=(1,1), D=(1,-1)) we have altogether 5 valleys strictly above the x-axis (indicated by |).

G. C. Greubel, Table of n, a(n) for n = 1..1000
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Cf. A119011.

R:=PowerSeriesRing(Rationals(), 30); [0,0] cat Coefficients(R!( 2*(1-3*x-(1-x)*Sqrt(1-4*x))/((1+2*x+ Sqrt(1-4*x))^2 *Sqrt(1-4*x)) )); // G. C. Greubel, Apr 06 2019

G:=2*(1-3*z-(1-z)*sqrt(1-4*z))/(1+2*z+sqrt(1-4*z))^2/sqrt(1-4*z): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=1..30);

Rest[CoefficientList[Series[2*(1-3*x-(1-x)*Sqrt[1-4*x])/(1+2*x+ Sqrt[1-4*x])^2/Sqrt[1-4*x], {x, 0, 30}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)

x='x+O('x^30); concat([0,0], Vec(2*(1-3*x-(1-x)*sqrt(1-4*x)) /( (1+2*x+sqrt(1-4*x))^2*sqrt(1-4*x)))) \\ G. C. Greubel, Mar 19 2017

a=(2*(1-3*x-(1-x)*sqrt(1-4*x)) /( (1+2*x+sqrt(1-4*x))^2* sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Apr 06 2019

a(n) = Sum_{k=0,..,n-2} k*A119011(n,k).
G.f.: 2*(1-3*z-(1-z)*sqrt(1-4*z))/((1+2*z+sqrt(1-4*z))^2*sqrt(1-4*z)).
a(n) ~ 2^(2*n+1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: 2*(n-3)*(3*n-2)*(n+2)*a(n) +(-21*n^3+50*n^2-13*n-4)*a(n-1) -2*(n-1) *(2*n-1)*(3*n+1)*a(n-2)=0. - R. J. Mathar, Jun 22 2016

A135336 Number of Dyck paths of semilength n with no UUDU's starting at level 0.

Original entry on oeis.org

1, 1, 2, 4, 10, 28, 85, 271, 893, 3013, 10351, 36075, 127219, 453097, 1627378, 5887660, 21436354, 78484402, 288779728, 1067263660, 3960081904, 14746806292, 55094725918, 206450572930, 775724723086, 2922060848734

Offset: 0

N. J. A. Sloane, Dec 07 2007

nonn

Column 0 of A135330. Partial sums of the Fine sequence 1,0,1,2,6,18,... (A000957 without the first term). - Emeric Deutsch, Dec 14 2007

			a(3)=4 because among the 5 (=A000108(3)) Dyck paths of semilength 3 only UUDUDD does not qualify.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Cf. A000108, A135330, A000957.

a:=proc(n) options operator, arrow: (sum((-1)^j*(3*j+1)*binomial(2*n-3*j,n), j =0..floor((1/3)*n)))/(n+1) end proc: seq(a(n),n=0..25); # Emeric Deutsch, Dec 14 2007

CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x)/((1-x)*(1+x*(1-Sqrt[1-4*x])/(2*x))), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

x='x+O('x^50); Vec((1-sqrt(1-4*x))/(x*(1-x)*(3 - sqrt(1-4*x)))) \\ G. C. Greubel, Mar 21 2017

From Emeric Deutsch, Dec 14 2007: (Start)
a(n) = Sum_{j=0..floor(n/3)} (-1)^j*(3*j+1)*binomial(2*n-3*j,n)/(n+1).
G.f.: C/(1+z^3*C^3) = C/[(1-z)*(1+z*C)], where C = [1-sqrt(1-4*z)]/(2*z) is the g.f. of the Catalan numbers (A000108). (End)
a(n) ~ 4^(n+2)/(27*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

Edited and extended by Emeric Deutsch, Dec 14 2007

A167769 Pendular trinomial triangle (p=0), read by rows of 2n+1 terms (n>=0), defined by the recurrence : if 0

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 6, 8, 6, 3, 1, 0, 0, 1, 4, 10, 18, 24, 18, 10, 4, 1, 0, 0, 1, 5, 15, 33, 57, 75, 57, 33, 15, 5, 1, 0, 0, 1, 6, 21, 54, 111, 186, 243, 186, 111, 54, 21, 6, 1, 0, 0, 1, 7, 28, 82, 193, 379, 622, 808, 622, 379, 193, 82, 28, 7, 1, 0, 0

Offset: 0

Philippe Deléham, Nov 11 2009

See A119369 for p=1 and A122445 for p=2. The diagonals may be generated by iterated convolutions of a base sequence B (A000108(n)) with the sequence C (A000957(n+1)) of central terms.

			Triangle begins :
  1;
  1, 0,  0;
  1, 1,  1,  0,  0;
  1, 2,  3,  2,  1,  0,  0;
  1, 3,  6,  8,  6,  3,  1,  0,  0;
  1, 4, 10, 18, 24, 18, 10,  4,  1, 0, 0,
  1, 5, 15, 33, 57, 75, 57, 33, 15, 5, 1, 0, 0; ...

Kim, Ki Hang; Rogers, Douglas G.; Roush, Fred W. Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577--594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013) - From N. J. A. Sloane, Jun 05 2012

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000108, A000957, A000958, A001558, A001559, A071724, A104629.

T:= proc(n, k) option remember;
      if k=0 and n=0 then 1;
    elif k<0 or k>2*(n-1) then 0;
    elif n=2 and k<3 then 1;
    elif kG. C. Greubel, Mar 17 2021

T[n_, k_]:= T[n, k]= If[k==0 && n==0, 1, If[k<0 || k>2*(n-1), 0, If[n==2 && k<3, 1, If[kG. C. Greubel, Mar 17 2021 *)

T(n, k)=if(k==0 && n==0, 1, if(k>2*n-2 || k<0, 0, if(n==2 && k<=2, 1, if(kPaul D. Hanna, Nov 12 2009

@CachedFunction
def T(n, k):
    if (k==0 and n==0): return 1
    elif (k<0 or k>2*(n-1)): return 0
    elif (n==2 and k<3): return 1
    elif (kG. C. Greubel, Mar 17 2021

Sum_{k=0..2*n} T(n,k) = A071724(n) = [n=0] + 3*binomial(2n,n-1)/(n+2) = [n=0] + n*C(n)/(n+2), where C(n) are the Catalan numbers (A000108). - G. C. Greubel, Mar 17 2021

A171368 Another version of A126216.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 5, 0, 9, 0, 1, 0, 0, 0, 21, 0, 14, 0, 1, 0, 0, 14, 0, 56, 0, 20, 0, 1, 0, 0, 0, 84, 0, 120, 0, 27, 0, 1, 0, 0, 42, 0, 300, 0, 225, 0, 35, 0, 1, 0, 0, 0, 330, 0, 825, 0, 385, 0, 44, 0, 1, 0, 0

Offset: 0

Philippe Deléham, Dec 06 2009

Expansion of the first column of the triangle T_(0,x), T_(x,y) defined in A039599; T_(0,0)= A053121, T_(0,1)= A089942, T_(0,2)= A126093, T_(0,3)= A126970.

T(n,k) is the number of Riordan paths of length n with k horizontal steps. A Riordan path is a Motzkin path with no horizontal steps on the x-axis. - Emanuele Munarini, Oct 14 2023

			Triangle begins:
  1 ;
  0,0 ;
  1,0,0 ;
  0,1,0,0 ;
  2,0,1,0,0 ;
  0,5,0,1,0,0 ;
  5,0,9,0,1,0,0 ;
  ...

Cf. A000108, A001003,

Sum_{k, 0<=k<=n} T(n,k)*x^k = A099323(n+1), A126120(n), A005043(n), A000957(n+1), A117641(n) for x = -1, 0, 1, 2, 3 respectively.

A237619 Riordan array (1/(1+xc(x)), xc(x)) where c(x) is the g.f. of Catalan numbers (A000108).

Original entry on oeis.org

1, -1, 1, 0, 0, 1, -1, 1, 1, 1, -2, 2, 3, 2, 1, -6, 6, 8, 6, 3, 1, -18, 18, 24, 18, 10, 4, 1, -57, 57, 75, 57, 33, 15, 5, 1, -186, 186, 243, 186, 111, 54, 21, 6, 1, -622, 622, 808, 622, 379, 193, 82, 28, 7, 1, -2120, 2120, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1

Offset: 0

Philippe Deléham, Feb 10 2014

			Triangle begins:
    1;
   -1,  1;
    0,  0,  1;
   -1,  1,  1,  1;
   -2,  2,  3,  2,  1;
   -6,  6,  8,  6,  3,  1;
  -18, 18, 24, 18, 10,  4, 1;
  -57, 57, 75, 57, 33, 15, 5, 1;
Production matrix begins:
  -1, 1;
  -1, 1, 1;
  -1, 1, 1, 1;
  -1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1, 1, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Diagonals: A000012, A000217, A023443, A166830.
Cf. A000957, A000958, A001558, A001559, A104629, A126983.
Cf. A065602, A167772.

A065602[n_, k_]:= A065602[n, k]= Sum[(k-1+2*j)*Binomial[2*(n-j)-k-1, n-1]/(2*(n - j) -k-1), {j,0,(n-k)/2}];
T[n_, k_]:= If[k==0, A065602[n, 0], If[n==1 && k==1, 1, A065602[n, k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 27 2022 *)

def A065602(n, k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
def A237619(n, k):
    if (n<2): return (-1)^(n-k)
    elif (k==0): return A065602(n, 0)
    else: return A065602(n, k)
flatten([[A237619(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2022

Sum_{k=0..n} T(n,k)*x^k = A126983(n), A000957(n+1), A026641(n) for x = 0, 1, 2 respectively.
T(n, k) = A167772(n-1, k-1) for k > 0, with T(n, 0) = A167772(n, 0).
T(n, 0) = A126983(n).
T(n+1, 1) = A000957(n+1).
T(n+2, 2) = A000958(n+1).
T(n+3, 3) = A104629(n) = A000957(n+3).
T(n+4, 4) = A001558(n).
T(n+5, 5) = A001559(n).
T(n, k) = A065602(n, k) for k > 0, with T(n, k) = (-1)^(n-k), for n < 2, and T(n, 0) = A065602(n, 0). - G. C. Greubel, May 27 2022

A253831 Number of 2-Motzkin paths with no level steps at height 1.

Original entry on oeis.org

1, 2, 5, 12, 30, 76, 197, 522, 1418, 3956, 11354, 33554, 102104, 319608, 1027237, 3381714, 11371366, 38946892, 135505958, 477781296, 1703671604, 6132978608, 22256615602, 81327116484, 298938112816, 1104473254912, 4098996843500, 15272792557230, 57106723430892, 214202598271360, 805743355591301

Offset: 0

José Luis Ramírez Ramírez, Apr 20 2015

nonn

For n=3 we have 12 paths: H(1)H(1)H(1), H(1)H(1)H(2), H(1)H(2)H(1), H(1)H(2)H(2), H(2)H(1)H(1), H(2)H(1)H(2), H(2)H(2)H(1), H(2)H(2)H(2), UDH(1), UDH(2), H(1)UD, H(2)UD.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A000957, A217312, A257363.

rec:= (54+36*n)*a(n)+(-3+7*n)*a(n+1)+(-60-36*n)*a(n+2)+(36+16*n)*a(n+3)+(-6-2*n)*a(n+4) = 0:
f:= gfun:-rectoproc({rec,seq(a(i)=[1,2,5,12][i+1],i=0..3)},a(n),remember):
seq(f(n),n=0..100); # Robert Israel, Apr 29 2015

CoefficientList[Series[1/(1-2*x-x*((1-Sqrt[1-4*x])/(3-Sqrt[1-4*x]))), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)

a(n):=sum(sum(((sum((k+1)*binomial(k+m,k+1)*binomial(2*j-k+m-1,j-k)*(-1)^(k),k,0,j))*2^(n-j-2*m)*binomial(n-m-j,m))/(j+m),j,0,n-2*m),m,1,n/2)+2^n; /* Vladimir Kruchinin, Mar 11 2016 */

G.f.: 1/(1-2*x-x*F(x)), where F(x) is the g.f. of Fine numbers A000957.
G.f.: 2*(2+x)/(4-7*x-6*x^2+x*sqrt(1-4*x)).
a(n) ~ 4^(n+1) / (25*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
(54+36*n)*a(n)+(-3+7*n)*a(n+1)+(-60-36*n)*a(n+2)+(36+16*n)*a(n+3)+(-6-2*n)*a(n+4) = 0. - Robert Israel, Apr 29 2015
a(n) = Sum_{m=1..n/2}(Sum_{j=0..n-2*m}(((Sum_{k=0..j}((k+1)*binomial(k+m,k+1)*binomial(2*j-k+m-1,j-k)*(-1)^(k)))*2^(n-j-2*m)*binomial(n-m-j,m))/(j+m)))+2^n. - Vladimir Kruchinin, Mar 11 2016

A294527 Number of Dyck paths of length 2n with exactly 2 hills.

Original entry on oeis.org

0, 0, 1, 0, 3, 6, 21, 66, 220, 744, 2562, 8942, 31569, 112530, 404445, 1464042, 5332872, 19532688, 71893470, 265778040, 986416614, 3674092044, 13729259586, 51455182260, 193369903608, 728504292576, 2750904025276, 10409856537786, 39470613237645, 149935171349546

Offset: 0

Eric M. Schmidt, Nov 01 2017

nonn

Dun Qiu and Jeffrey B. Remmel, Quadrant marked mesh patterns in 123-avoiding permutations, arXiv:1705.00164 [math.CO], 2017, p. 28.

Column k=2 of A065600. Cf. A000957, A065601.

a[n_] := Which[n>2, Sum[(i Binomial[i+2, i] Binomial[2n-2i-4, n-2])/(n-i-2), {i, 0, (n-2)/2}], n == 2, 1, True, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 27 2018 *)

Conjecture: 2*(3*n-5) *(n-2) *(3*n+11) *(n+1) *a(n) -(3*n+11) *(n-3) *(21*n^2-35*n+10) *a(n-1) -2*(3*n+11) *(n-1) *(2*n-1) *(3*n-2) *a(n-2)= 0. - R. J. Mathar, Jun 24 2018

cp's OEIS Frontend

A059027 Number of Dyck paths of semilength n with no peak at height 4.

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A105640 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k UUDD's, where U=(1,1) and D=(1,-1) (0<=k<=floor(n/2), n>=2). A hill in a Dyck path is a peak at level 1.

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A118974 Sum of the lengths of the first descents in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1).

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A119012 Number of valleys strictly above the x-axis in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1).

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A135336 Number of Dyck paths of semilength n with no UUDU's starting at level 0.

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A167769 Pendular trinomial triangle (p=0), read by rows of 2n+1 terms (n>=0), defined by the recurrence : if 0

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A171368 Another version of A126216.

A237619 Riordan array (1/(1+xc(x)), xc(x)) where c(x) is the g.f. of Catalan numbers (A000108).