A005802 -id:A005802 - cp's OEIS frontend

A000139 a(n) = 2(3n)! / ((2n+1)!(n+1)!).

2, 1, 2, 6, 22, 91, 408, 1938, 9614, 49335, 260130, 1402440, 7702632, 42975796, 243035536, 1390594458, 8038677054, 46892282815, 275750636070, 1633292229030, 9737153323590, 58392041019795, 352044769046880, 2132866978427640, 12980019040145352, 79319075627675556

Offset: 0

N. J. A. Sloane, entry revised Apr 24 2012

This sequence arises in many different contexts, and over the years it has had several different definitions. I have now changed the definition back to one of the earlier ones, a self-contained formula. - N. J. A. Sloane, Apr 24 2012
The number of 2-stack sortable permutations on n letters (n >= 1).
The number of rooted non-separable planar maps with n+1 edges. - Valery A. Liskovets, Mar 17 2005
The shifted sequence starting with a(1): Number of quadrangular dissections of a square, counted by the number of vertices. Rooted, non-separable planar maps with no multiple edges, in which each non-root face has degree 4.
Number of left ternary trees having n nodes (n>=1). - Emeric Deutsch, Jul 23 2006
A combinatorial interpretation for this sequence in terms of a family of plane trees is given in [Schaeffer, Corollary 2 with k = 3]. - Peter Bala, Oct 12 2011
Number of canopy intervals in the Tamari lattices, see [Préville-Ratelle and Viennot, section 6]. - F. Chapoton, Apr 19 2015
The number of fighting fish (branching polyominoes). - David Bevan, Jan 10 2018
The number of 1324-avoiding dominoes (gridded permutations). - David Bevan, Jan 10 2018
For n > 0, a(n) is the number of simple strong triangulations of a fixed quadrilateral with n interior nodes. See A210664. - Andrew Howroyd, Feb 24 2021
Conjecture: a(n) is odd iff n is a term of A022341. - Peter Bala, Jul 24 2025

			G.f. = 2 + x + 2*x^2 + 6*x^3 + 22*x^4 + 91*x^5 + 408*x^6 + 1938*x^7 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 365.
Eric S. Egge, Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, pp. 65-82 of "A Century of Advancing Mathematics", ed. S. F. Kennedy et al., MAA Press 2015.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. See p. 399 Table A.7
W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.41.

T. D. Noe, Table of n, a(n) for n = 0..200
A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011. See p. 15.
E. Ben-Naim and P. L. Krapivsky, Popularity-Driven Networking, arXiv preprint arXiv:1112.0049 [cond-mat.stat-mech], 2011.
Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See pp. 10-11.
David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, arXiv preprint arXiv:1711.10325 [math.CO], 2017.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
M. Bousquet-Mélou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, arXiv:math/0504018 [math.CO], 2005; J Comb. Thy. B 96 (2006), 623-672.
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.
Colin Defant, Counting 3-Stack-Sortable Permutations, arXiv:1903.09138 [math.CO], 2019.
A. Del Lungo, F. Del Ristoro and J.-G. Penaud, Left ternary trees and non-separable rooted planar maps, Theor. Comp. Sci., 233, 2000, 201-215.
E. Duchi, V. Guerrini, S. Rinaldi and G. Schaeffer, Fighting fish: enumerative properties, Sém. Lothar. Combin. 78B (2017), Art. 43, 12 pp.
E. Duchi, V. Guerrini, S. Rinaldi, and G. Schaeffer, Fighting fish, J. Phys. A, Math. Theor. 50, No. 2, Article ID 024002, 16 p. (2017).
Enrica Duchi and Corentin Henriet, A bijection between rooted planar maps and generalized fighting fish, arXiv:2210.16635 [math.CO], 2022.
S. Dulucq, S. Gire, and O. Guibert, A combinatorial proof of J. West's conjecture Discrete Math. 187 (1998), no. 1-3, 71--96. MR1630680(99f:05053).
S. Dulucq, S. Gire, and J. West, Permutations with forbidden subsequences and nonseparable planar maps, Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993). Discrete Math. 153 (1996), no. 1-3, 85-103. MR1394948 (98a:05081)
Eric S. Egge, Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, MAA Focus, August/September 2015, pp. 33-34. [Annotated scanned copy]
W. Fang, Fighting fish and two-stack sortable permutations, arXiv preprint arXiv:1711.05713 [math.CO], 2017.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 713
I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, Electron. J Combin. 13 (2006), paper 53.
Juan B. Gil, Oscar A. Lopez, and Michael D. Weiner, A positional statistic for 1324-avoiding permutations, arXiv:2311.18227 [math.CO], 2023.
O. Guibert, Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps, Discr. Math., 210 (2000), 71-85.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
S. Kitaev, P. Salimov, C. Severs and H. Ulfarsson, Restricted non-separable planar maps and some pattern avoiding permutations, 2012.
S. Kitaev, P. Salimov, C. Severs and H. Ulfarsson, Restricted non-separable planar maps and some pattern avoiding permutations, Discrete Applied Mathematics, Volume 161, Issues 16-17, November 2013, Pages 2514-2526.
Sergey Kitaev, Anna de Mier, and Marc Noy, On the number of self-dual rooted maps, European J. Combin. 35 (2014), 377-387. MR3090510. See Theorem 1.
Sergey Kitaev, Pavel Salimov, Christopher Severs, and Henning Ulfarsson, Restricted non-separable planar maps and some pattern avoiding permutations, arXiv preprint arXiv:1202.1790 [math.CO], 2012. [Original title: Restricted rooted non-separable planar maps]
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Alois Panholzer, Parking function varieties for combinatorial tree models, arXiv:2007.14676 [math.CO], 2020.
Permutation Pattern Avoidance Library (PermPAL), 1324 Domino
L.-F. Préville-Ratelle and X. Viennot, An extension of Tamari lattices, arXiv preprint arXiv:1406.3787 [math.CO], 2014.
G. Schaeffer, A combinatorial interpretation of super-Catalan numbers of order two, (2001).
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.
J. West, Sorting twice through a stack, Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991), Theoret. Comput. Sci. 117 (1993), no. 1-2, 303-313.
D. Zeilberger, A proof of Julian West's conjecture that the number of two-stacksortable permutations of length n is 2(3n)!/((n + 1)!(2n + 1)!), Discrete Math., 102 (1992), 85-93.
P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the counting of tangles and links, Discrete Math 246 (2002), 343-360.
P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, arXiv:math-ph/0303049, 2003; J. Knot Theor. Ramifications 13 (2004) 325-356.

Cf. A000142, A000309, A001764, A005802, A006335, A004677, A121545, A218473, A233389.

Row m=1 of A210664(n>0).

a000139 0 = 2
a000139 n = ((3 * n) `a007318` (2 * n + 1)) `div` a000217 n
-- Reinhard Zumkeller, Feb 17 2013

[2*Factorial(3*n)/(Factorial(2*n+1)*Factorial(n+1)): n in [0..25]]; // Vincenzo Librandi, Apr 20 2015

A000139 := n->2*(3*n)!/((2*n+1)!*((n+1)!)): seq(A000139(n), n=0..23);

Table[(2(3n)!)/((2n+1)!(n+1)!),{n,0,30}] (* Harvey P. Dale, Mar 31 2013 *)

a(n)=binomial(3*n,n)*2/((n+1)*(2*n+1)); \\ Joerg Arndt, Jul 21 2014

from sympy import binomial
def A000139(n): return (binomial(3*n, n)*2)//((n+1)*(2*n+1))

A000139_list = [2]
for n in range(1,30):
    A000139_list.append(3*(3*n-2)*(3*n-1)*A000139_list[-1]//(2*n+2)//(2*n+1)) # Chai Wah Wu, Apr 02 2021

def A000139(n): return (binomial(3*n, n)*2)//((n+1)*(2*n+1))
[A000139(n) for n in (0..23)]  # Peter Luschny, Jun 17 2013

a(n) = 2*binomial(3*n, 2*n+1)/(n*(n+1)), or 2*(3*n)!/((2*n+1)!*((n+1)!)).
Using Stirling's formula in A000142 it is easy to get the asymptotic expression a(n) ~ (27/4)^n / (sqrt(Pi*n / 3) * (2*n + 1) * (n + 1)). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 13 2001
G.f.: A(z) = 2 + z*B(z), where B(z) = 1 - 8*z + 2*z*(5-6*z)*B - 2*z^2*(1+3*z)*B^2 - z^4*B^3.
G.f.: (2/(3*x)) * (hypergeom([-2/3, -1/3],[1/2],(27/4)*x)-1). - Mark van Hoeij, Nov 02 2009
G.f.: (2-3*R)/(R-1)^2 where R := RootOf(x-t*(t-1)^2,t) is an algebraic function in Maple notation. - Mark van Hoeij, Nov 08 2011
G.f.: 2*Q(0), where Q(k) = 1 + 3*x*(3*k+1)*(6*k+1)/(2*(k+1)*(4*k+3) - 6*x*(k+1)*(3*k+2)*(4*k+3)*(6*k+5)/(3*x*(3*k+2)*(6*k+5) + (2*k+3)*(4*k+5)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
E.g.f.: 2*Q(0), where Q(k) = 1 + 3*x*(3*k+1)*(6*k+1)/(2*(k+1)*(2*k+1)*(4*k+3) - 6*x*(k+1)*(2*k+1)*(3*k+2)*(4*k+3)*(6*k+5)/(3*x*(3*k+2)*(6*k+5) + (2*k+2)*(2*k+3)*(4*k+5)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
a(n) = A007318(3*n, 2*n+1)/A000217(n) for n > 0. - Reinhard Zumkeller, Feb 17 2013
a(n) is the n-th Hausdorff moment of the positive function w(x) defined on (0,27) which is equal to w(x) = 3*sqrt(3)*2^(2/3)*(3-sqrt(81-12*x)/9)*(1+sqrt(81-12*x)/9)^(1/3)/(8*Pi*x^(2/3))-sqrt(3)*2^(1/3)*(3+sqrt(81-12*x)/9)*(1+sqrt(81-12*x)/9)^(-1/3)/(4*Pi*x^(1/3)), that is, a(n) is the integral Integral_{x=0..27/4} x^n*w(x) dx, n >= 0. The function w(x) is unique. - Karol A. Penson, Jun 17 2013
D-finite with recurrence 2*(n+1)*(2*n+1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Aug 21 2014
G.f. A(z) is related to the g.f. M(z) of A000168 by M(z) = 1 + A(z*M(z)^2) (see Tutte 1963, equation 6.3). - Noam Zeilberger, Nov 02 2016
From Ilya Gutkovskiy, Jan 17 2017: (Start)
E.g.f.: 2*2F2(1/3,2/3; 3/2,2; 27*x/4).
Sum_{n>=0} 1/a(n) = (1/2)*3F2(1,3/2,2; 1/3,2/3; 4/27) = 2.226206199291261... (End)
G.f. A(z) is the solution to the initial value problem 4*A + 2*z*A' = 8 + 3*z*A + 9*z^2*A' + 2*z^2*A*A', A(0) = 2. - Bjarki Ágúst Guðmundsson, Jul 03 2017
a(n+1) = a(n)*3*(3*n+1)*(3*n+2)/(2*(n+2)*(2*n+3)). - Chai Wah Wu, Apr 02 2021
a(n) = 4*(3*n)!/(n!*(2*n+2)!). - Chai Wah Wu, Dec 15 2021
From Peter Bala, Feb 05 2022: (Start)
O.g.f.: A(x) = T(x)*(3 - T(x)), where T(x) = 1 + x*T(x)^3 is the o.g.f. of A001764.
(1/x)*(A(x) - 2)/(A(x) - 1) = 1 + x + 3*x^2 + 11*x^3 + 46*x^4 + 209*x^5 + ... is the o.g.f. of A233389.
1 + 2*x*A'(2*x)/A(2*x) = 1 + x + 7*x^2 + 61*x^3 + 591*x^4 + 6101*x^6 + ... is the o.g.f. of A218473.
Let B(x) = 1 + x*(A(x) - 1). Then x*B'(x)/B(x) = x + x^2 + 4*x^3 + 17*x^4 + 81*x^5 + ... is the o.g.f. of A121545. (End)

A047889 Number of permutations in S_n with longest increasing subsequence of length <= 4.

Original entry on oeis.org

1, 1, 2, 6, 24, 119, 694, 4582, 33324, 261808, 2190688, 19318688, 178108704, 1705985883, 16891621166, 172188608886, 1801013405436, 19274897768196, 210573149141896, 2343553478425816, 26525044132374656, 304856947930144656

Offset: 0

Eric Rains (rains(AT)caltech.edu), N. J. A. Sloane

Or, number of permutations in S_n that avoid the pattern 12345, - N. J. A. Sloane, Mar 19 2015

Also, the dimension of the space of SL(4)-invariants in V^m ⊗ (V^*)^m, where V is the standard 4-dimensional representation of SL(4) and V^* its dual. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

			G.f. = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 119*x^5 + 694*x^6 + 4582*x^7 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..300
F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513 [math.CO], 2015.
Peter J. Forrester and Fei Wei, Higher order linear differential equations for unitary matrix integrals: applications and generalisations, arXiv:2508.20797 [math-ph], 2025.
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53 (1990), 257-285.
Permutation Pattern Avoidance Library (PermPAL), 12345
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
Index entries for sequences related to Young tableaux.

A column of A047888.
Cf. A005802, A047890, A052399.
Column k=4 of A214015.
Representatives for the 16 Wilf-equivalence patterns of length 5 are given in A116485, A047889, and A256195-A256208. - N. J. A. Sloane, Mar 19 2015

A:=rsolve({a(0) = 1, a(1) = 1, (n^3 + 16*n^2 + 85*n + 150)*a(n + 2) = (20*n^3 + 182*n^2 + 510*n + 428)*a(n + 1) - (64*n^3 + 256*n^2 + 320*n +128)*a(n)}, a(n), makeproc): # Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

Flatten[{1,RecurrenceTable[{64*(-1+n)^2*n*a[-2+n]-2*(-12 + 11*n + 31*n^2 + 10*n^3)*a[-1+n] + (3+n)^2*(4+n)*a[n]==0,a[1]==1,a[2]==2},a,{n,20}]}] (* Vaclav Kotesovec, Sep 10 2014 *)

{a(n) = my(v); if( n<2, n>=0, v = vector(n+1, k, 1); for(k=2, n, v[k+1] = ((20*k^3 + 62*k^2 + 22*k - 24) * v[k] - 64*k*(k-1)^2 * v[k-1]) / ((k+3)^2 * (k+4))); v[n+1])}; /* Michael Somos, Apr 19 2015 */

a(0)=1, a(1)=1, (n^3 + 16*n^2 + 85*n + 150)*a(n+2) = (20*n^3 + 182*n^2 + 510*n + 428)*a(n+1) - (64*n^3 + 256*n^2 + 320*n + 128)*a(n). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
a(n) = (64*(n+1)*(2*n^3 + 21*n^2 + 76*n + 89)*A002895(n) -(8*n^4 + 104*n^3 + 526*n^2 + 1098*n + 776) *A002895(n+1)) / (3*(n+2)^3*(n+3)^3*(n+4)). - Mark van Hoeij, Jun 02 2010
a(n) ~ 3 * 2^(4*n + 9) / (n^(15/2) * Pi^(3/2)). - Vaclav Kotesovec, Sep 10 2014

More terms from Naohiro Nomoto, Mar 01 2002

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

A214015 Number of permutations A(n,k) in S_n with longest increasing subsequence of length <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 5, 1, 0, 1, 1, 2, 6, 14, 1, 0, 1, 1, 2, 6, 23, 42, 1, 0, 1, 1, 2, 6, 24, 103, 132, 1, 0, 1, 1, 2, 6, 24, 119, 513, 429, 1, 0, 1, 1, 2, 6, 24, 120, 694, 2761, 1430, 1, 0, 1, 1, 2, 6, 24, 120, 719, 4582, 15767, 4862, 1, 0

Offset: 0

Alois P. Heinz, Jul 01 2012

A(n,k) is also the sum of the squares of numbers of standard Young tableaux (SYT) of height <= k over all partitions of n.
This array is a larger and reflected version of A047888.
Column k>1 is asymptotic to (Product_{j=1..k} j!) * k^(2*n + k^2/2) / (Pi^((k-1)/2) * 2^((k-1)*(k+2)/2) * n^((k^2-1)/2)). - Vaclav Kotesovec, Sep 10 2014

			A(4,2) = 14 because 14 permutations of {1,2,3,4} do not contain an increasing subsequence of length > 2: 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312, 4321.  Permutation 1423 is not counted because it contains the noncontiguous increasing subsequence 123.
A(4,2) = 14 = 2^2 + 3^2 + 1^2 because the partitions of 4 with <= 2 parts are [2,2], [3,1], [4] with 2, 3, 1 standard Young tableaux, respectively:
  +------+  +------+  +---------+  +---------+  +---------+  +------------+
  | 1  3 |  | 1  2 |  | 1  3  4 |  | 1  2  4 |  | 1  2  3 |  | 1  2  3  4 |
  | 2  4 |  | 3  4 |  | 2 .-----+  | 3 .-----+  | 4 .-----+  +------------+
  +------+  +------+  +---+        +---+        +---+
Square array A(n,k) begins:
  1,  1,   1,    1,    1,    1,    1,    1, ...
  0,  1,   1,    1,    1,    1,    1,    1, ...
  0,  1,   2,    2,    2,    2,    2,    2, ...
  0,  1,   5,    6,    6,    6,    6,    6, ...
  0,  1,  14,   23,   24,   24,   24,   24, ...
  0,  1,  42,  103,  119,  120,  120,  120, ...
  0,  1, 132,  513,  694,  719,  720,  720, ...
  0,  1, 429, 2761, 4582, 5003, 5039, 5040, ...

Alois P. Heinz, Antidiagonals n = 0..70, flattened
Wikipedia, Longest increasing subsequence problem
Wikipedia, Young tableau

Columns k=0-10 give: A000007, A000012, A000108, A005802, A047889, A047890, A052399, A072131, A072132, A072133, A072167.
Differences between A000142 and columns k=0-9 give: A000142 (for n>0), A033312, A056986, A158005, A158432, A159139, A159175, A217675, A217676, A217677.
Main diagonal and first lower diagonal give: A000142, A033312.
A(2n,n-1) gives A269042(n) for n>0.
Cf. A047887, A047888, A182172, A208447, A214152, A267479.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
                 add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
A:= (n, k)-> `if`(k>=n, n!, g(n, k, [])):
seq(seq(A(n, d-n), n=0..d), d=0..14);

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]! / Product[Product[1+l[[i]]-j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]];
A[n_, k_] := If[k >= n, n!, g[n, k, {}]];
Table [Table [A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

A099952 Number of Wilf classes in S_n.

Original entry on oeis.org

1, 1, 1, 3, 16, 91, 595

Offset: 1

N. J. A. Sloane, Nov 12 2004

Z. Stankova and J. West, A new class of Wilf-equivalent permutations, J. Algeb. Combin., 15 (2002), 271-290.

A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.
A. M. Baxter and A. D. Jaggard, Pattern avoidance by even permutations, arXiv preprint arXiv:1106.3653, 2011
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152. See Fig. 9.

Representatives for the three Wilf classes in S_4 are A005802, A022558, A061552. - N. J. A. Sloane, Mar 15 2015

Representatives for the 16 Wilf-equivalence patterns of length 5 are given in A116485, A047889, and A256195-A256208. - N. J. A. Sloane, Mar 19 2015

A022558 Number of permutations of length n avoiding the pattern 1342.

Original entry on oeis.org

1, 1, 2, 6, 23, 103, 512, 2740, 15485, 91245, 555662, 3475090, 22214707, 144640291, 956560748, 6411521056, 43478151737, 297864793993, 2059159989914, 14350039389022, 100726680316559, 711630547589023, 5057282786190872, 36132861123763276, 259423620328055093

Offset: 0

Miklos Bona

Differs from A117156 which counts permutations avoiding the *consecutive* pattern 1342. - Ray Chandler, Dec 06 2011
Also, number of permutation of length n avoiding the pattern 3142 (see Stankova (1994) below). - Alexander Burstein, Aug 09 2013
Conjecture: a(n) is the number of permutations of length n simultaneously avoiding patterns 2143 and 415263. - Alexander Burstein, Mar 21 2019

			a(4) = 23 because obviously all permutations of length 4 with the exception of 1342 avoid 1342.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 768, Th. 12.1.14.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.48.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Miklos Bona, Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps, arXiv:math/9702223 [math.CO], 1997.
Miklos Bona, Exact enumeration of 1342-avoiding permutations; A close link with labeled trees and planar maps, J. Combinatorial Theory, A80 (1997), 257-272.
Alexander Burstein and Jay Pantone, Two examples of unbalanced Wilf-equivalence, J. Combin. 6 (2015), no. 1-2, 55-67.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
A. R. Conway and A. J. Guttmann, On 1324-avoiding permutations, Adv. Appl. Math. 64 (2015), 50-69.
A. L. L. Gao, S. Kitaev, and P. B. Zhang. On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
C. Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint arXiv:1410.2657 [math.CO], 2014.
Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
W. Mlotkowski, K. A. Penson, A Fuss-type family of positive definite sequences, arXiv:1507.07312 (2015), eq. (36).
Z. E. Stankova, Forbidden subsequences, Discrete Math., 132 (1994), no. 1-3, 291-316.
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001. See Fig. 11.

Essentially the same as A004040.
Cf. A117158.
A005802, A022558, A061552 are representatives for the three Wilf classes for length-four avoiding permutations (cf. A099952).

a := proc (n) options operator, arrow: (1/2)*(-1)^(n-1)*(7*n^2-3*n-2)+3*(sum((-1)^(n-i)*2^(i+1)*factorial(2*i-4)*binomial(n-i+2, 2)/(factorial(i)*factorial(i-2)), i = 2 .. n)) end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Oct 15 2014

Table[SeriesCoefficient[32*x/(1+20*x-8*x^2-(1-8*x)^(3/2)),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 07 2012 *)
Table[1/2*(-1)^(n-1) * (-2-3*n+7*n^2) + 1/4*(-1)^n * (1+n) * (-2-13*n+(n+2) * Hypergeometric2F1[-3/2,-n,-2-n,-8]),{n,0,20}] (* Vaclav Kotesovec, Aug 24 2014 *)

x='x+O('x^66); Vec( 32*x/(1+20*x-8*x^2-(1-8*x)^(3/2)) ) \\ Joerg Arndt, May 04 2013

a(n) = (7*n^2-3*n-2)/2 * (-1)^(n-1) + 3*Sum_{i=2..n} 2^(i+1) * (2*i-4)!/(i!*(i-2)!) * binomial(n-i+2, 2) * (-1)^(n-i).
G.f.: 32*x/(1 + 20*x - 8*x^2 - (1 - 8*x)^(3/2)). - Emeric Deutsch, Mar 13 2004
Recurrence: n*a(n) = (7*n-22)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 07 2012
a(n) ~ 2^(3*n+6)/(243*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Oct 07 2012

Minor edits by Vaclav Kotesovec, Aug 24 2014

A039963 The period-doubling sequence A035263 repeated.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane

nonn

An example of a d-perfect sequence.
Motzkin numbers mod 2. - Benoit Cloitre, Mar 23 2004
Let {a, b, c, c, a, b, a, b, a, b, c, c, a, b, ...} be the fixed point of the morphism: a -> ab, b -> cc, c -> ab, starting from a; then the sequence is obtained by taking a = 1, b = 1, c = 0. - Philippe Deléham, Mar 28 2004
The asymptotic mean of this sequence is 2/3 (Rowland and Yassawi, 2015; Burns, 2016). - Amiram Eldar, Jan 30 2021
The Gilbreath transform of floor(log_2(n)) (A000523). - Thomas Scheuerle, Sep 02 2024

Seiichi Manyama, Table of n, a(n) for n = 0..10000
T. Amdeberhan and M. Alekseyev, A moment sequence and Motzkin numbers. Modular coincidence?, MathOverflow, 2021.
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.
David Kohel, San Ling and Chaoping Xing, Explicit Sequence Expansions, in: C. Ding, T. Helleseth and H. Niederreiter (eds.), Sequences and their Applications, Proceedings of SETA'98 (Singapore, 1998), Discrete Mathematics and Theoretical Computer Science, 1999, pp. 308-317; alternative link.
Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, Journal de Théorie des Nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288; arXiv preprint, arXiv:1310.8635 [math.NT], 2013-2014.

Cf. A000523, A005802, A081706.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

Flatten[ Nest[ Function[l, {Flatten[(l /. {a -> {a, b}, b -> {c, c}, c -> {a, b}})]}], {a}, 7] /. {a -> {1}, b -> {1}, c -> {0}}] (* Robert G. Wilson v, Feb 26 2005 *)

A039963(n) = 1 - valuation(n\2+1,2)%2; \\ Max Alekseyev, Oct 23 2021

def A039963(n): return ((m:=(n>>1)+1)&-m).bit_length()&1 # Chai Wah Wu, Jan 09 2023

a(n) = A035263(1+floor(n/2)). - Benoit Cloitre, Mar 23 2004
a(n) = A040039(n) mod 2 = A002212(n+1) mod 2. a(0) = a(1) = 1, for n>=2: a(n) = ( a(n) + Sum_{k=0..n-2} a(k)*a(n-2-k)) mod 2. - Philippe Deléham, Mar 26 2004
a(n) = (A(n+2) - A(n)) mod 2, for A = A019300, A001285, A010060, A010059, A000069, A001969. - Philippe Deléham, Mar 28 2004
a(n) = A001006(n) mod 2. - Christian G. Bower, Jun 12 2005
a(n) = (-1)^n*(A096268(n+1) - A096268(n)). - Johannes W. Meijer, Feb 02 2013
a(n) = 1 - A007814(floor(n/2)+1) mod 2 = A005802(n) mod 2. - Max Alekseyev, Oct 23 2021

More terms from Christian G. Bower, Jun 12 2005

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe and Ralf Stephan, Jul 13 2007

A061552 Number of 1324-avoiding permutations of length n.

Original entry on oeis.org

1, 1, 2, 6, 23, 103, 513, 2762, 15793, 94776, 591950, 3824112, 25431452, 173453058, 1209639642, 8604450011, 62300851632, 458374397312, 3421888118907, 25887131596018, 198244731603623, 1535346218316422, 12015325816028313, 94944352095728825, 757046484552152932, 6087537591051072864

Offset: 0

Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001

nonn

			a(4)=23 because all 24 permutations of length 4, except 1324 itself, avoid the pattern 1324.

Miklós Bóna, Combinatorics of Permutations. Discrete Mathematics and its Applications (Boca Raton), 2nd edn. CRC Press, Boca Raton (2012).

David Bevan, Table of n, a(n) for n = 0..50 (from the Conway/Guttmann reference; terms 0..31 by Joerg Arndt, taken from the Johansson/Nakamura reference; terms 37..50 by Bjarki Ágúst Guðmundsson, taken from the Conway/Guttmann/Zinn-Justin reference).
M. H. Albert, M. Elder, A. Rechnitzer, P. Westcott, and M. Zabrocki, On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia, arXiv:math/0502504 [math.CO], 2005.
R. Arratia, On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern., Electron. J. Combin. 6, N1 (1999).
David Bevan, Permutations avoiding 1324 and patterns in Łukasiewicz paths, arXiv:1406.2890 [math.CO], 2014-2015.
David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, European J. Combin. 88 (2020).
Miklós Bóna, A new upper bound for 1324-avoiding permutations, arXiv:1207.2379 [math.CO], 2012.
Miklós Bóna, A new upper bound for 1324-avoiding permutations, Combin. Probab. Comput. 23(5), 717-724 (2014).
Miklós Bóna, A new record for 1324-avoiding permutations, arXiv:1404.4033 [math.CO], 2014.
Miklós Bóna, A new record for 1324-avoiding permutations, European Journal of Mathematics (2015) 1:198-206, DOI 10.1007/s40879-014-0020-6.
A. Claesson, V. Jelínek, and E. Steingrímsson, Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns. J. Combin. Theory Ser. A 119(8), 1680-1691 (2012).
Andrew R. Conway and Anthony J. Guttmann, On the growth rate of 1324-avoiding permutations, arXiv:1405.6802 [math.CO], (2014).
Andrew R. Conway, Anthony J. Guttmann and Paul Zinn-Justin, 1324-avoiding permutations revisited, arXiv preprint arXiv:1709.01248 [math.CO], 2017.
Steven Finch, Pattern-Avoiding Permutations [Broken link?]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]
A. L. L. Gao, S. Kitaev, and P. B. Zhang, On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
Ruth Hoffmann, Özgür Akgün, and Christopher Jefferson, Composable Constraint Models for Permutation Enumeration, arXiv:2311.17581 [cs.DM], 2023.
Fredrik Johansson and Brian Nakamura, Using functional equations to enumerate 1324-avoiding permutations, arXiv:1309.7117 [math.CO], (2013).
A. Marcus and G. Tardos, Excluded permutation matrices and the Stanley-Wilf conjecture, J. Combin. Theory Ser. A 107(1), 153-160 (2004).
B. K. Nakamura, Computational methods in permutation patterns, PhD Dissertation, Rutgers University, May 2013.
Brian Nakamura and Doron Zeilberger, Using Noonan-Zeilberger Functional Equations to enumerate (in Polynomial Time!) Generalized Wilf classes, arXiv preprint arXiv:1209.2353 [math.CO], 2012.
Anthony Zaleski and Doron Zeilberger, On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts, arXiv:1712.10072 [math.CO], 2017.

A005802, A022558, A061552 are representatives for the three Wilf classes for length-four avoiding permutations (cf. A099952).

count1324 := proc(n::nonnegint) if (n<4) then return n!; fi; if (n=4) then return 23; fi; return nodes([5,5,5,5], n-5) + nodes([5,3,5,5], n-5) + nodes([5,4,4,5], n-5) + nodes([5,5,4,5], n-5) + nodes([4,3,4], n-5) + nodes([5,3,4,5], n-5); end:
nodes := proc(p, h) option remember; local i, j, s, l; if (h=0) then return convert(p, `+`); fi; s := 0; for j to nops(p) do l := p[j]+1; for i from 2 to j do l := l, `min`(j+1, p[i]); od; for i from j+1 to p[j] do l := l, p[i-1]+1; od; s := s+nodes([l], h-1); od; return s; end:

a[n_] := n!/;n<4; a[4]=23; a[n_] := Total[nodes[#,n-5]&/@{{4,3,4},{5,3,4,5},{5,3,5,5},{5,4,4,5},{5,5,4,5},{5,5,5,5}}]; nodes[p_,0]:=Total[p]; nodes[p_,h_] := nodes[p,h] = Sum[nodes[Join[{p[[j]]+1}, Min[j+1,#]&/@p[[2;;j]], p[[j;;p[[j]]-1]]+1],h-1], {j,Length[p]}]; Array[a,12] (* David Bevan, May 25 2012 *)

More terms from Vincent Vatter, Feb 26 2005

a(23)-a(25) added from the Albert et al. paper by N. J. A. Sloane, Mar 29 2013

A047890 Number of permutations in S_n with longest increasing subsequence of length <= 5.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 719, 5003, 39429, 344837, 3291590, 33835114, 370531683, 4285711539, 51990339068, 657723056000, 8636422912277, 117241501095189, 1639974912709122, 23570308719710838, 347217077020664880, 5231433025400049936, 80466744544235325387

Offset: 0

Eric Rains (rains(AT)caltech.edu), N. J. A. Sloane

Alois P. Heinz, Table of n, a(n) for n = 0..735
F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513, 2015.
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
Permutation Pattern Avoidance Library (PermPAL), 012345
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
Index entries for sequences related to Young tableaux.

A column of A047888. Cf. A005802, A052399.

Column k=5 of A214015.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l)
      `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
       add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n, 5, []):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 10 2012
# second Maple program
a:= proc(n) option remember; `if`(n<6, n!, ((-375+400*n+843*n^2
       +322*n^3+35*n^4)*a(n-1) +225*(n-1)^2*(n-2)^2*a(n-3)
       -(259*n^2+622*n+45)*(n-1)^2*a(n-2))/ ((n+6)^2*(n+4)^2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 26 2012

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_, k_] := If[k >= n, n!, g[n, k, {}]]; Table[a[n, 5], {n, 1, 30}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

a(n) ~ 9 * 5^(2*n + 25/2) / (512 * n^12 * Pi^2). - Vaclav Kotesovec, Sep 10 2014

More terms from Naohiro Nomoto, Mar 01 2002

More terms from Alois P. Heinz, Apr 10 2012

A052399 Number of permutations in S_n with longest increasing subsequence of length <= 6.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5039, 40270, 361302, 3587916, 38957991, 457647966, 5763075506, 77182248916, 1091842643475, 16219884281650, 251774983140578, 4066273930979460, 68077194367392864, 1177729684507324152, 20995515989327134152, 384762410996641402384

Offset: 0

N. J. A. Sloane, Mar 13 2000

nonn

Previous name was: Related to Young tableaux of bounded height.

Alois P. Heinz, Table of n, a(n) for n = 0..250
F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513 [math.CO], 2015.
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
Index entries for sequences related to Young tableaux.

Cf. A005802, A047889, A047890.

Column k=6 of A214015.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
       +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) option remember;
      `if`(n=0, h(l)^2, `if`(i<1, 0, `if`(i=1, h([l[], 1$n])^2,
       g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
a:= n-> g(n, 6, []):
seq(a(n), n=0..25); # Alois P. Heinz, Apr 10 2012
# second Maple program
a:= proc(n) option remember; `if`(n<7, n!,
      ((56*n^5-9408+11032*n+19028*n^2+7360*n^3+1092*n^4)*a(n-1)
       -4*(196*n^3+1608*n^2+3167*n+444)*(n-1)^2*a(n-2)
       +1152*(2*n+3)*(n-1)^2*(n-2)^2*a(n-3))/ ((n+9)*(n+8)^2*(n+5)^2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 26 2012

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_, k_] := If[k >= n, n!, g[n, k, {}]]; Table[a[n, 6], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)

a(n) ~ 5 * 2^(2*n + 6) * 3^(2*n + 21) / (n^(35/2) * Pi^(5/2)). - Vaclav Kotesovec, Sep 10 2014

More terms from Alois P. Heinz, Apr 10 2012

New name from Vaclav Kotesovec, Sep 10 2014

cp's OEIS Frontend

A158005 Numbers of pattern-matching permutations of (1234) for the permutations of {1, 2, ..., n} on n = 4, 5, 6, ... elements.

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A000139 a(n) = 2*(3*n)! / ((2*n+1)!*(n+1)!).

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A047889 Number of permutations in S_n with longest increasing subsequence of length <= 4.

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A214015 Number of permutations A(n,k) in S_n with longest increasing subsequence of length <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A099952 Number of Wilf classes in S_n.

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A022558 Number of permutations of length n avoiding the pattern 1342.

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A039963 The period-doubling sequence A035263 repeated.

A000139 a(n) = 2(3n)! / ((2n+1)!(n+1)!).