A000957 -id:A000957 - cp's OEIS frontend

A114590 Number of peaks at even levels in all hill-free Dyck paths of semilength n+2 (a hill in a Dyck path is a peak at level 1).

1, 2, 8, 28, 103, 382, 1432, 5408, 20546, 78436, 300636, 1156188, 4459267, 17241526, 66807856, 259361920, 1008598126, 3928120924, 15319329472, 59817190552, 233826979750, 914962032172, 3583556424208, 14047386554368, 55108441878868

Offset: 0

Emeric Deutsch, Dec 11 2005

nonn

			a(1)=2 because in the 2 (=A000957(4)) hill-free Dyck paths of semilength 3, namely UUUDDD and U(UD)(UD)D (U=(1,1), D=(1,-1)) we have altogether 2 peaks at even level (shown between parentheses).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A114588, A114587, A114515.

G:=(1+2*z^2-(1+2*z)*sqrt(1-4*z))/2/z^2/(2+z)^2/sqrt(1-4*z): Gser:=series(G,z=0,30): 1, seq(coeff(Gser,z^n),n=1..25);

CoefficientList[Series[(1+2*x^2-(1+2*x)*Sqrt[1-4*x])/2/x^2/(2+x)^2/Sqrt[1-4*x], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

G.f.: (1+2*z^2-(1+2*z)*sqrt(1-4*z))/(2*z^2*(2+z)^2*sqrt(1-4*z)).
a(n) = sum(k*A114588(n+2,k),k=0..n+1).
a(n)=sum{k=0..n, sum{j=0..n-k, C(n-j,k-j)*C(n-j,k)*(j+1)}}; - Paul Barry, Nov 03 2006
Conjecture: 2*(n+2)*a(n) +(-7*n-9)*a(n-1) -18*a(n-2) +2*(-7*n+19)*a(n-3) +4*(-2*n+3)*a(n-4)=0. - R. J. Mathar, Nov 15 2012
Recurrence: 2*n*(n+2)*(3*n+1)*a(n) = (21*n^3 + 34*n^2 + n - 8)*a(n-1) + 2*(n+1)*(2*n+1)*(3*n+4)*a(n-2). - Vaclav Kotesovec, Feb 12 2014
a(n) ~ 4^(n+2) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 12 2014

A118973 Number of hill-free Dyck paths of semilength n+2 and having length of first descent equal to 2 (a hill in a Dyck path is a peak at level 1).

Original entry on oeis.org

1, 0, 2, 5, 16, 51, 168, 565, 1934, 6716, 23604, 83806, 300154, 1083137, 3934404, 14374413, 52787766, 194746632, 721435884, 2682522918, 10008240456, 37455101382, 140569122624, 528926230530, 1994980278636, 7541234323096

Offset: 0

Emeric Deutsch, May 08 2006

nonn

Also, for a given j>=2, number of hill-free Dyck paths of semilength n+j and having length of first descent equal to j. a(n)=A000108(n+1)-A000108(n)-[A000957(n+2)-A000957(n+1)]. Columns 2,3,4,... of A118972 (without the initial 0's).

			a(2)=2 because we have uu(dd)uudd and uuu(dd)udd, where u=(1,1),d=(1,-1) (the first descents are shown between parentheses).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Cf. A000108, A000957, A118972, A001558.

F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=(1-z)*C*F: gser:=series(g,z=0,33): seq(coeff(gser,z,n),n=0..28);
A118973List := proc(m) local A, P, n; A := [1,0]; P := [1,0];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-2]]);
A := [op(A), P[-1]] od; A end: A118973List(26); # Peter Luschny, Mar 26 2022

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

a(n):=(sum((k+2)*(-1)^k*(binomial(2*n-k+1,n-k)/(n+2)-binomial(2*n-k-1,n-k-1)/(n+1)),k,0,n-1))+(-1)^(n); /* Vladimir Kruchinin, Mar 06 2016 */

my(x='x+O('x^25)); Vec((1-x)*(1-sqrt(1-4*x))/x/(3-sqrt(1-4*x))*(1-sqrt(1-4*x))/2/x) \\ G. C. Greubel, Feb 08 2017

G.f.: (1-x)*C*F, where F = (1-sqrt(1-4*x))/(x*(3-sqrt(1-4*x))) and C = (1-sqrt(1-4*x))/(2*x) is the Catalan function.
a(n) ~ 5*4^n/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
a(n) = (Sum_{k=0..n-1}((k+2)*(-1)^k*(binomial(2*n-k+1,n-k)/(n+2)-binomial(2*n-k-1,n-k-1)/(n+1))))+(-1)^(n). - Vladimir Kruchinin. Mar 06 2016
D-finite with recurrence +2*(n+2)*a(n) +(-7*n-2)*a(n-1) +2*(-3*n+1)*a(n-2) +(7*n-26)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

A167772 Riordan array (c(x)/(1+xc(x)), xc(x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 6, 8, 6, 3, 1, 18, 24, 18, 10, 4, 1, 57, 75, 57, 33, 15, 5, 1, 186, 243, 186, 111, 54, 21, 6, 1, 622, 808, 622, 379, 193, 82, 28, 7, 1, 2120, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1, 7338, 9458, 7338, 4596, 2476, 1164, 474, 163, 45, 9, 1

Offset: 0

Philippe Deléham, Nov 11 2009, corrected Nov 12 2009

			Triangle begins:
     1;
     0,    1;
     1,    1,    1;
     2,    3,    2,    1;
     6,    8,    6,    3,   1;
    18,   24,   18,   10,   4,   1;
    57,   75,   57,   33,  15,   5,   1;
   186,  243,  186,  111,  54,  21,   6,  1;
   622,  808,  622,  379, 193,  82,  28,  7,  1;
  2120, 2742, 2120, 1312, 690, 311, 118, 36,  8,  1;
Production matrix begins:
  0, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  ... - _Philippe Deléham_, Mar 03 2013

Reinhard Zumkeller, Rows n=0..125 of triangle, flattened

Cf. A000957, A000958 (row sums), A001558, A001559, A104629, A167769.

Diagonals: A000012, A000217, A001477, A166830.

import Data.List (genericIndex)
a167772 n k = genericIndex (a167772_row n) k
a167772_row n = genericIndex a167772_tabl n
a167772_tabl = [1] : [0, 1] :
               map (\xs@(:x:) -> x : xs) (tail a065602_tabl)
-- Reinhard Zumkeller, May 15 2014

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+1,3] + Boole[n==0], A065602[n+1, k+1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 26 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 A167772(n,k):
    if (k==0): return A065602(n+1,3) + bool(n==0)
    else: return A065602(n+1,k+1)
flatten([[A167772(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2022

Sum_{k=0..n} T(n, k) = A000958(n+1).
From Philippe Deléham, Nov 12 2009: (Start)
Sum_{k=0..n} T(n,k)*2^k = A014300(n).
Sum_{k=0..n} T(n,k)*2^(n-k) = A064306(n). (End)
For n > 0: T(n,0) = A065602(n+1,3), T(n,k) = A065602(n+1,k+1), k = 1..n. - Reinhard Zumkeller, May 15 2014

A114586 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at odd levels (0<=k<=n-2; n>=2). A hill in a Dyck path is a peak at level 1.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 6, 8, 3, 1, 15, 22, 15, 4, 1, 36, 68, 52, 24, 5, 1, 91, 198, 191, 100, 35, 6, 1, 232, 586, 651, 425, 170, 48, 7, 1, 603, 1718, 2203, 1656, 820, 266, 63, 8, 1, 1585, 5047, 7285, 6299, 3591, 1435, 392, 80, 9, 1, 4213, 14808, 23832, 23164, 15155, 6972, 2338

Offset: 2

Emeric Deutsch, Dec 11 2005

Row sums are the Fine numbers (A000957). Column 0 yield the Riordan numbers (A005043). Sum(k*T(n,k),k=0..n-2)=A114587(n).

			T(5,2)=3 because we have UU(UD)DU(UD)DD, UUDU(UD)(UD)DD and UU(UD)(UD)DUDD, where U=(1,1), D=(1,-1) (the peaks at odd levels are shown between parentheses).
Triangle begins:
1;
1,1;
3,2,1;
6,8,3,1;
15,22,15,4,1;

Cf. A000957, A005043, A114587, A114588, A100754.

G:=(t*z+z+1-sqrt(z^2*t^2+2*z^2*t-2*z*t-3*z^2-2*z+1))/2/z/(1+t+z)-1: Gser:=simplify(series(G,z=0,15)): for n from 2 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 2 to 12 do seq(coeff(t*P[n],t^j),j=1..n-1) od; # yields sequence in triangular form

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

A114588 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at even levels (0<=k<=n-1; n>=1). A hill in a Dyck path is a peak at level 1.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 3, 1, 1, 3, 6, 6, 2, 1, 7, 17, 18, 11, 3, 1, 17, 48, 58, 40, 18, 4, 1, 43, 134, 186, 150, 76, 27, 5, 1, 110, 380, 590, 540, 325, 130, 38, 6, 1, 286, 1083, 1860, 1915, 1305, 624, 206, 51, 7, 1, 753, 3100, 5844, 6660, 5115, 2772, 1097, 308, 66, 8, 1, 2003

Offset: 1

Emeric Deutsch, Dec 11 2005

Row n has n terms. Row sums are the Fine numbers (A000957). Column 0 yields A114589. Sum(k*T(n,k), k=0..n-1) yields A114590.

			T(4,3) = 1 because we have U(UD)(UD)(UD)D, where U=(1,1), D=(1,-1) (the peaks at even levels are shown between parentheses).
Triangle begins:
0;
0,   1;
1,   0,  1;
1,   3,  1,  1;
3,   6,  6,  2,  1;
7,  17, 18, 11,  3, 1;
17, 48, 58, 40, 18, 4, 1;

Cf. A000957, A114586, A114587, A114589, A114590.

G:=(1-t*z+2*z^2+3*z-2*t*z^2-sqrt(1-3*z^2-2*z*t+2*z^2*t+z^2*t^2-2*z))/2/z/(2+2*z-t*z-t*z^2+z^2)-1: Gser:=simplify(series(G, z=0, 15)): for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 12 do seq(coeff(t*P[n], t^j), j=1..n) od; # yields sequence in triangular form

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

A116914 Number of UUDD's, where U=(1,1) and D=(1,-1), 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

1, 1, 5, 16, 58, 211, 781, 2920, 11006, 41746, 159154, 609324, 2341060, 9021559, 34855741, 134972368, 523689718, 2035462990, 7923732118, 30889008112, 120566373676, 471134916286, 1842964183570, 7216096752496, 28279240308268, 110913181145716, 435333520075796, 1709861650762900

Offset: 2

Emeric Deutsch, May 08 2006

nonn

Catalan transform of A034299. - R. J. Mathar, Jun 29 2009

			a(4)=5 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UU(UUDD)DD, UUUDUDDD, UUD(UUDD)D, UUDUDUDD, U(UUDD)UDD and (UUDD)(UUDD) (U=(1,1), D=(1,-1)) we have altogether 5 UUDD's (shown between parentheses).

G. C. Greubel, Table of n, a(n) for n = 2..1000
David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, Y. Vaughan, Pattern Avoiding Linear Extensions of Rectangular Posets, arXiv:1605.06825 [math.CO], 2016.
Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Cf. A105640.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( x*(1 + 5*x-(1-x)*Sqrt(1-4*x))/(2*(2+x)^2*Sqrt(1-4*x)) )); // G. C. Greubel, May 08 2019

G:=z*(1+5*z-(1-z)*sqrt(1-4*z))/2/(2+z)^2/sqrt(1-4*z): Gser:=series(G,z=0,31): seq(coeff(Gser,z^n),n=2..28);

Rest[Rest[CoefficientList[Series[x*(1+5*x-(1-x)*Sqrt[1-4*x])/2/(2+x)^2/Sqrt[1-4*x], {x, 0, 40}], x]]] (* Vaclav Kotesovec, Mar 20 2014 *)

my(x='x+O('x^40)); Vec(x*(1+5*x-(1-x)*sqrt(1-4*x))/(2*(2+x)^2 *sqrt(1-4*x))) \\ G. C. Greubel, Feb 08 2017

a=(x*(1+5*x-(1-x)*sqrt(1-4*x))/(2*(2+x)^2*sqrt(1-4*x))).series(x, 40).coefficients(x, sparse=False); a[2:] # G. C. Greubel, May 08 2019

a(n) = Sum_{k=0..floor(n/2)} k*A105640(n,k).
G.f.: x*(1 + 5*x - (1-x)*sqrt(1-4*x))/(2*(2+x)^2*sqrt(1-4*x)).
a(n+2) = A126258(2*n,n). - Philippe Deléham, Mar 13 2007
a(n) ~ 2^(2*n-1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence +2*(-n+1)*a(n) +3*(-n+6)*a(n-1) +3*(13*n-44)*a(n-2) +10*(2*n-5)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

A118972 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having length of first descent equal to k (1<=k<=n; n>=1). A hill in a Dyck path is a peak at level 1.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 3, 2, 0, 1, 10, 5, 2, 0, 1, 33, 16, 5, 2, 0, 1, 111, 51, 16, 5, 2, 0, 1, 379, 168, 51, 16, 5, 2, 0, 1, 1312, 565, 168, 51, 16, 5, 2, 0, 1, 4596, 1934, 565, 168, 51, 16, 5, 2, 0, 1, 16266, 6716, 1934, 565, 168, 51, 16, 5, 2, 0, 1, 58082, 23604, 6716, 1934, 565, 168

Offset: 1

Emeric Deutsch, May 08 2006

Row sums are the Fine numbers (A000957).
T(n,1) = A001558(n-3) for n>=3.
T(n,k) = A118973(n-k) for n>=k>=2.
Sum_{k=1..n} k*T(n,k) = A118974(n).

			T(5,2)=5 because we have uu(dd)uududd, uu(dd)uuuddd,uuu(dd)uuddd,uuu(dd)ududd and uuuu(dd)uddd, where u=(1,1), d=(1,-1) (the first descents are shown between parentheses).
Triangle starts:
  0;
  0,1;
  1,0,1;
  3,2,0,1;
  10,5,2,0,1;
  33,16,5,2,0,1;
  ...

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

Cf. A000957, A001558, A118973, A118974.

F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: G:=t*z^2*C*F*(C-(1-t)/(1-t*z)): Gser:=simplify(series(G,z=0,15)): for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 1 to 12 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form

G.f.: t*z^2*C*F*(C-(1-t)/(1-t*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.

A122920 Diagonal sums of number triangle A122919.

Original entry on oeis.org

1, 1, 4, 12, 39, 129, 436, 1498, 5218, 18386, 65420, 234734, 848403, 3086001, 11288412, 41499354, 153247278, 568188606, 2114334312, 7893906144, 29561195238, 111007927386, 417918303144, 1577061975492, 5964172347604, 22601012748124, 85806694043116, 326343785428946, 1243200250005995

Offset: 0

Paul Barry, Sep 19 2006

Starting with offset 1 = iterates of M * [1,1,1,0,0,0,...] where M is the tridiagonal matrix with [0,2,2,2,...] as the main diagonal and [1,1,1,...] as the super and subdiagonals. - Gary W. Adamson, Jan 09 2009

Partial sums are Fine numbers (A000957) with offset 3. - Alexander Burstein, Apr 15 2015

			G.f. = 1 + x + 4*x^2 + 12*x^3 + 39*x^4 + 129*x^5 + 436*x^6 + 1498*x^7 + 5218*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.

Cf. A000957.

CoefficientList[Series[((1-x)*(1-2*x-2*x^2-Sqrt[1-4*x])/(2*(2+x)*x^3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 03 2014 *)
Table[9/(16 (-2)^n) + 3 (2n+4)! HypergeometricPFQ[{1, n+5/2, n+3}, {n+2, n+5}, -8]/((n+1)! (n+4)!), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 26 2015 *)

x='x+O('x^66); Vec(((1-x)*(1-2*x-2*x^2-sqrt(1-4*x))/(2*(2+x)*x^3))) \\ Joerg Arndt, May 08 2013

G.f.: ((1-x)*(1-2*x-2*x^2-sqrt(1-4*x))/(2*(2+x)*x^3)).
Conjecture: 2*n*(n+3)*a(n) - (7*n^2+9*n+4)*a(n-1) - 2*(n+1)*(2*n+1)*a(n-2) = 0. - R. J. Mathar, Nov 05 2012
a(n) ~ 2^(2*n+4) / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 03 2014
From Vladimir Reshetnikov, Oct 26 2015: (Start)
a(n) = 9/(16*(-2)^n) + 3*(2*n+4)!*hypergeom([1,n+5/2,n+3], [n+2,n+5], -8)/((n+1)!*(n+4)!).
a(n) = 9/(16*(-2)^n) + 8*2^n*(2*n+5)!!*hypergeom([1,n+7/2], [n+5], -8)/(n+4)! - 4*2^n*(2*n+3)!!*hypergeom([1,n+5/2], [n+4], -8)/(n+3)!. (End)
G.f. A(x) =: y satisfies 0 = (1 - x)^2 - y*(1 - 3*x + 2*x^3) + y^2*(2*x^3 + x^4). - Michael Somos, Oct 26 2015
0 = a(n)*(+16*a(n+1) - 26*a(n+2) - 98*a(n+3) + 36*a(n+4)) + a(n+1)*(+50*a(n+1) + 35*a(n+2) - 179*a(n+3) + 46*a(n+4)) + a(n+2)*(+105*a(n+2) + 47*a(n+3) - 50*a(n+4)) + a(n+3)*(+14*a(n+3) + 4*a(n+4)) for all n>=0. - Michael Somos, Oct 26 2015

A306409 a(n) = -Sum_{0<=i

Original entry on oeis.org

0, 1, 3, 10, 34, 120, 434, 1597, 5949, 22363, 84655, 322245, 1232205, 4729453, 18210279, 70307546, 272087770, 1055139408, 4099200524, 15951053566, 62159391150, 242542955378, 947504851414, 3705431067156, 14505084243860, 56831711106496, 222853334131080

Offset: 0

Seiichi Manyama, Apr 05 2019

nonn

			n | a(n) | A307354 | A006134 | A120305
--+------+---------+---------+---------
0 |    0 |       1 |       1 |       1
1 |    1 |       2 |       3 |       1
2 |    3 |       6 |       9 |       3
3 |   10 |      19 |      29 |       9
4 |   34 |      65 |      99 |      31
5 |  120 |     231 |     351 |     111

Seiichi Manyama, Table of n, a(n) for n = 0..1664

Partial sums of A014300. - Seiichi Manyama, Jan 30 2023

Cf. A000108, A000957, A006134, A057552, A120305, A307354.

Table[-Sum[Sum[(-1)^(i+j) * (i+j)!/(i!*j!), {i, 0, j-1}], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 05 2019 *)

a(n) = -sum(i=0, n, sum(j=i+1, n, (-1)^(i+j)*(i+j)!/(i!*j!)));

my(N=30, x='x+O('x^N)); concat(0, Vec((1-sqrt(1-4*x))/(sqrt(1-4*x)*(1-x)*(3-sqrt(1-4*x))))) \\ Seiichi Manyama, Jan 30 2023

a(n) = A006134(n) - A307354(n).
a(n) = (A006134(n) - A120305(n))/2.
a(n) ~ 4^(n+1) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 05 2019
G.f.: ( 1/(sqrt(1-4*x) * (1-x)) ) * ( x *c(x)/(1 + x *c(x)) ), where c(x) is the g.f. of A000108. - Seiichi Manyama, Jan 30 2023

A309508 Number of cyclic permutations of length n avoiding the pattern 321.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 24, 66, 178, 512, 1486, 4446, 13468, 41648, 130178, 412670, 1321418, 4274970, 13948966, 45890440, 152061154, 507292698, 1702753462, 5748085332, 19506240462

Offset: 0

Miklos Bona, Aug 05 2019

Comment from F. Chapoton, Sep 14 2021: (Start)
The maps sending a permutation to its inverse or to its reverse-complement define two commuting involutions on these sets of permutations.
The next terms in the sequence could be 41648, 130178, though these are counting Dyck words such that an associated permutation is cyclic, related but not obviously equivalent combinatorial objects. (End)

			For n=3, there are two such permutations, 231 and 312.
The a(4) = 4 permutations are: 2341, 2413, 3142, 4123.
The a(5) = 10 permutations are: 23451, 23514, 24153, 25134, 31452, 31524, 34512, 41253, 45123, 51234.

Kassie Archer, Christina Graves, and Robert Laudone, Binary operations on pattern-avoiding cycles, arXiv:2505.04456 [math.CO], 2025.
Miklos Bona and Michael Cory, Cyclic Permutations Avoiding Pairs of Patterns of Length Three, arXiv:1805.05196 [math.CO], 2018.

Cf. A000108 (number of permutations avoiding 321).

Cf. A000957, A309504, A309505, A309506.

\\ See PARI link in A309504 for program code.
for(n=0, 16, print1(E321(n), ", ")) \\ Andrew Howroyd, Nov 20 2024

a(0)=1 prepended and a(13)-a(24) from Andrew Howroyd, Nov 17 2024

cp's OEIS Frontend

A114590 Number of peaks at even levels in all hill-free Dyck paths of semilength n+2 (a hill in a Dyck path is a peak at level 1).

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A118973 Number of hill-free Dyck paths of semilength n+2 and having length of first descent equal to 2 (a hill in a Dyck path is a peak at level 1).

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A167772 Riordan array (c(x)/(1+x*c(x)), x*c(x)), c(x) the g.f. of A000108.

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A114586 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at odd levels (0<=k<=n-2; n>=2). A hill in a Dyck path is a peak at level 1.

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A114588 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at even levels (0<=k<=n-1; n>=1). A hill in a Dyck path is a peak at level 1.

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A116914 Number of UUDD's, where U=(1,1) and D=(1,-1), 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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A118972 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having length of first descent equal to k (1<=k<=n; n>=1). A hill in a Dyck path is a peak at level 1.

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A167772 Riordan array (c(x)/(1+xc(x)), xc(x)), c(x) the g.f. of A000108.