A069270 -id:A069270 - cp's OEIS frontend

A069269 Second level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 3rd level is A069270).

1, 1, 1, 1, 2, 3, 1, 3, 7, 12, 1, 4, 12, 30, 55, 1, 5, 18, 55, 143, 273, 1, 6, 25, 88, 273, 728, 1428, 1, 7, 33, 130, 455, 1428, 3876, 7752, 1, 8, 42, 182, 700, 2448, 7752, 21318, 43263, 1, 9, 52, 245, 1020, 3876, 13566, 43263, 120175, 246675

Offset: 0

Henry Bottomley, Mar 12 2002

For the m-th level generalization of Catalan triangle T(n,k) = C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n >= k+m: T(n,k) = T(n-m+1,k+1) - T(n-m,k+1); and T(n,n) = T(n+m-1,n-1) = C((m+1)n,n)/(mn+1).
Reflected version of A110616. - Philippe Deléham, Jun 15 2007
With offset 1 for n and k, T(n,k) is (conjecturally) the number of permutations of [n] that avoid the patterns 4-2-3-1 and 4-2-5-1-3 and for which the last ascent ends at position k (k=1 if there are no ascents). For example, T(4,1) = 1 counts 4321; T(4,2) = 3 counts 1432, 2431, 3421; T(4,3) = 7 counts 1243, 1342, 2143, 2341, 3142, 3241, 4132. - David Callan, Jul 22 2008
Row sums appear to be in A098746. - R. J. Mathar, May 30 2014

			Rows start
  1;
  1,  1;
  1,  2,  3;
  1,  3,  7, 12;
  1,  4, 12, 30, 55;

M. H. Albert et al., Restricted permutations and queue jumping, Discrete Math., 287 (2004), 129-133.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 18.

Columns include A000012, A000027, A055998.
Right-hand diagonals include A001764, A006013, A006629, A006630, A006631.
Cf. triangles A007318, A009766, A069270.

T(n, k) = C(n+2k, k)*(n-k+1)/(n+k+1).
For n >= k+2: T(n, k) = T(n-1, k+1) - T(n-2, k+1).
T(n, n) = T(n+1, n-1) = C(3n, n)/(2n+1).

A069271 a(n) = binomial(4n+1,n)2/(3*n+2).

Original entry on oeis.org

1, 2, 9, 52, 340, 2394, 17710, 135720, 1068012, 8579560, 70068713, 580034052, 4855986044, 41043559340, 349756577100, 3001701610320, 25921837477692, 225083787458904, 1963988670706228, 17211860478150800, 151433425446423120

Offset: 0

Henry Bottomley, Mar 12 2002

nonn

This sequence counts the set B_n of plane trees defined in the Poulalhon and Schaeffer link (Definition 2.2 and Section 4.2, Proposition 4). - David Callan, Aug 20 2014
a(n) is the number of lattice paths of length 4n starting and ending on the x-axis consisting of steps {(1, 1), (1, -3)} that remain on or above the line y=-1. - Sarah Selkirk, Mar 31 2020
a(n) is the number of ordered pairs of 4-ary trees with a (summed) total of n internal nodes. - Sarah Selkirk, Mar 31 2020

			a(3) = C(4*3+1,3)*2/(3*3+2) = C(13,3)*2/11 = 286*2/11 = 52.
a(3) = 52 since the top row of M^3 = (22, 22, 7, 1).
1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 + 135720*x^7 + ...
q + 2*q^3 + 9*q^5 + 52*q^7 + 340*q^9 + 2394*q^11 + 17710*q^13 + 135720*q^15 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Thomas Anderson, T. Bruce McLean, Homeira Pajoohesh, and Chasen Smith , The combinatorics of all regular flexagons, Eu. J. Combinat. 31 (2010) 72-80, Theorem 2.
Roland Bacher, Fair Triangulations, arXiv:0710.0960 [math.CO], 2007.
Esther Banaian, Elise Catania, Christian Gaetz, Miranda Moore, Gregg Musiker, and Kayla Wright, Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras, arXiv:2507.15211 [math.CO], 2025. See p. 28.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Gi-Sang Cheon, S.-T. Jin and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.
Emmanuel Guitter, The distance-dependent two-point function of triangulations: a new derivation from old results, Ann. Inst. Henri Poincaré Comb. Phys. Interact. Vol. 4 (2017), 177-211. DOI: 10.4171/AIHPD/38.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Ionut E. Iacob, T. Bruce McLean and Hua Wang, The V-flex, Triangle Orientation, and Catalan Numbers in Hexaflexagons, The College Mathematics Journal, Vol. 43, No. 1 (January 2012), pp. 6-10.
Jean-Christophe Novelli and Jean-Yves Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.
C. O. Oakley and R. J. Wisner, Flexagons, Am. Math. Monthly 64 (3) (1957) 143-154, u_{3k+2}.
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv:1103.3453 [math-ph], 2011.
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, Phys. Rev. E 83, 061118, 2011.
Dominique Poulalhon and Gilles Schaeffer, Optimal Coding and Sampling of Triangulations, in Automata, Languages and Programming, Lecture Notes in Computer Science, Volume 2719, 2003, pp 1080-1094.
Sarah J. Selkirk, On a generalisation of k-Dyck paths, MSc Thesis, 2019.

Cf. A002293, A006013, A006632, A069270 for similar generalized Catalan sequences.

Cf. A000260, A115141, A163456, A224274.

[2*Binomial(4*n+1, n)/(3*n+2): n in [0..20]];  // Bruno Berselli, Mar 04 2011

BB:=[T,{T=Prod(Z,Z,Z,F,F),F=Sequence(B),B=Prod(F,F,F,Z)}, unlabeled]: seq(count(BB,size=i),i=3..23); # Zerinvary Lajos, Apr 22 2007

f[n_] := 2 Binomial[4 n + 1, n]/(3 n + 2); Array[f, 21, 0] (* Robert G. Wilson v *)

a(n)=if(n<0,0,polcoeff(serreverse(x/(1+x^2)^2+O(x^(2*n+2))),2*n+1)) /* Ralf Stephan */

{a(n) =  binomial(4*n + 2, n)*2 / (2*n + 1)} /* Michael Somos, Mar 28 2012 */

{a(n) =  local(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = (1 + x * A^2)^2); polcoeff( A, n))} /* Michael Somos, Mar 28 2012 */

a(n) = A069270(n+1, n) = A005810(n)*A016813(n)/A060544(n+1)
O.g.f. A(x) satisfies 2*x^2*A(x)^3 = 1-2*x*A(x)-sqrt(1-4*x*A(x)). - Vladimir Kruchinin, Feb 23 2011
a(n) is the sum of top row terms in M^n, where M is the infinite square production matrix with the triangular series in each column as follows, with the rest zeros:
   1,  1,  0, 0, 0, 0, ...
   3,  3,  1, 0, 0, 0, ...
   6,  6,  3, 1, 0, 0, ...
  10, 10,  6, 3, 1, 0, ...
  15, 15, 10, 6, 3, 1, ...
  ... - Gary W. Adamson, Aug 11 2011
Given g.f. A(x) then B(x) = x * A(x^2) satisfies x = B(x) / (1 + B(x)^2)^2. - Michael Somos, Mar 28 2012
Given g.f. A(x) then A(x) = (1 + x * A(x)^2)^2. - Michael Somos, Mar 28 2012
a(n) / (n+1) = A000260(n). - Michael Somos, Mar 28 2012
REVERT transform is A115141. - Michael Somos, Mar 28 2012
D-finite with recurrence 3*n*(3*n+2)*(3*n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Jun 07 2013
a(n) = 2*binomial(4n+1,n-1)/n for n>0, a(0)=1. - Bruno Berselli, Jan 19 2014
G.f.: hypergeom([1/2, 3/4, 5/4], [4/3, 5/3], (256/27)*x). - Robert Israel, Aug 24 2014
From Peter Bala, Oct 08 2015: (Start)
O.g.f. A(x) = (1/x) * series reversion (x/C(x)^2), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. Cf. A163456.
(1/2)*x*A'(x)/A(x) is the o.g.f. for A224274. (End)
E.g.f.: hypergeom([1/2, 3/4, 5/4], [1, 4/3, 5/3], (256/27)*x). - Karol A. Penson, Jun 26 2017
a(n) = binomial(4*n+2,n)/(2*n+1). - Alexander Burstein, Nov 08 2021

A156064 Inverse of Riordan array (1/(1-x^4), x/(1-x^4)), A156062.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, -2, 0, 0, 0, 1, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, -4, 0, 0, 0, 1, 4, 0, 0, 0, -5, 0, 0, 0, 1, 0, 9, 0, 0, 0, -6, 0, 0, 0, 1, 0, 0, 15, 0, 0, 0, -7, 0, 0, 0, 1, 0, 0, 0, 22, 0, 0, 0, -8, 0, 0, 0, 1, -22, 0, 0, 0, 30, 0, 0, 0, -9, 0, 0, 0, 1, 0, -52, 0, 0, 0, 39

Offset: 0

Paul Barry, Oct 20 2009

Reverse and aerate A069270. First column is signed aerated version of A002293. Diagonal sums are A156065.

			Triangle begins
    1;
    0,  1;
    0,  0,  1;
    0,  0,  0,  1;
   -1,  0,  0,  0,  1;
    0, -2,  0,  0,  0,  1;
    0,  0, -3,  0,  0,  0,  1;
    0,  0,  0, -4,  0,  0,  0,  1;
    4,  0,  0,  0, -5,  0,  0,  0,  1;
    0,  9,  0,  0,  0, -6,  0,  0,  0,  1;
    0,  0, 15,  0,  0,  0, -7,  0,  0,  0,  1;
    0,  0,  0, 22,  0,  0,  0, -8,  0,  0,  0,  1;
  -22,  0,  0,  0, 30,  0,  0,  0, -9,  0,  0,  0,  1;
Production matrix is
   0,  1;
   0,  0,  1;
   0,  0,  0,  1;
  -1,  0,  0,  0,  1;
   0, -1,  0,  0,  0,  1;
   0,  0, -1,  0,  0,  0,  1;
   0,  0,  0, -1,  0,  0,  0,  1;
   0,  0,  0,  0, -1,  0,  0,  0,  1;
   0,  0,  0,  0,  0, -1,  0,  0,  0,  1;
   0,  0,  0,  0,  0,  0, -1,  0,  0,  0,  1;
   0,  0,  0,  0,  0,  0,  0, -1,  0,  0,  0,  1;
   0,  0,  0,  0,  0,  0,  0,  0, -1,  0,  0,  0,  1;
   0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  0,  0,  0,  1;
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  0,  0,  0,  1;

Chai Wah Wu, Record values in appending and prepending bitstrings to runs of binary digits, arXiv:1810.02293 [math.NT], 2018.

cp's OEIS Frontend

A069269 Second level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 3rd level is A069270).

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A069271 a(n) = binomial(4n+1,n)2/(3*n+2).

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A156064 Inverse of Riordan array (1/(1-x^4), x/(1-x^4)), A156062.

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cp's OEIS Frontend

A069269 Second level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 3rd level is A069270).

Views

Author

Keywords

Comments

Examples

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Crossrefs

Formula

A069271 a(n) = binomial(4*n+1,n)*2/(3*n+2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

Formula

A156064 Inverse of Riordan array (1/(1-x^4), x/(1-x^4)), A156062.

Views

Author

Keywords

Comments

Examples

Links

A069271 a(n) = binomial(4n+1,n)2/(3*n+2).