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A069271 a(n) = binomial(4*n+1,n)*2/(3*n+2). 21
1, 2, 9, 52, 340, 2394, 17710, 135720, 1068012, 8579560, 70068713, 580034052, 4855986044, 41043559340, 349756577100, 3001701610320, 25921837477692, 225083787458904, 1963988670706228, 17211860478150800, 151433425446423120 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

T. Anderson, T. B. McLean, H. Pajoohesh, C. 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.

Gi-Sang Cheon, S.-T. Jin, 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.

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.

J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.

C. O. Oakley, 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.

FORMULA

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

O.g.f. A(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. - Peter Bala, Oct 08 2015

E.g.f.: hypergeom([1/2, 3/4, 5/4], [1, 4/3, 5/3], (256/27)*x). - Karol A. Penson, Jun 26 2017

EXAMPLE

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 + ...

MAPLE

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

MATHEMATICA

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

PROG

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

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

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

(PARI) {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 */

CROSSREFS

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

Cf. A000260, A115141, A163456, A224274.

Sequence in context: A193465 A003584 A301928 * A305987 A231494 A006152

Adjacent sequences:  A069268 A069269 A069270 * A069272 A069273 A069274

KEYWORD

nonn

AUTHOR

Henry Bottomley, Mar 12 2002

STATUS

approved

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Last modified May 6 15:28 EDT 2021. Contains 343586 sequences. (Running on oeis4.)