A211216 Expansion of (1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16795, 58766, 207783, 740924, 2660139, 9603089, 34818270, 126676726, 462125928, 1689438278, 6186432967, 22682699779, 83249302471, 305773834030, 1123771473120, 4131947428007, 15197952958467, 55915691993228

Offset: 0

Author

Bruno Berselli, May 11 2012

Keywords

• nonn
• easy

Comments

In the paper of Kitaev, Remmel and Tiefenbruck (see the Links section), Q_(132)^(k,0,0,0)(x,0) represents a generating function depending on k and x.

For successive values of k we have:

k=1, the g.f. of A000012: 1/(1-x);

k=2, " A011782: (1-x)/(1-2*x);

k=3, " A001519: (1-2*x)/(1-3*x+x^2);

k=4, " A124302: (1-3*x+x^2)/(1-4*x+3*x^2);

k=5, " A080937: (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3);

k=6, " A024175: (1-5*x+6*x^2-x^3)/(1-6*x+10*x^2-4*x^3);

k=7, " A080938: (1-6*x+10*x^2-4*x^3)/(1-7*x+15*x^2-10*x^3+x^4);

k=8, " A033191: (1-7*x+15*x^2-10*x^3+x^4)/(1-8*x+21*x^2

-20*x^3+5*x^4).

This sequence corresponds to the case k=9.

We observe that the coefficients of numerators and denominators are in A115139.

In general, Q_(132)^(k,0,0,0)(x,0) is the generating function for Dyck paths whose maximum height is less than or equal to k; also, it is the generating function of rooted binary trees T which have no nodes 'eta' such that there are >= k left edges on the path from 'eta' to the root of T (see cited paper, page 11).

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (page 10, Corollary 3).
• K. Mészáros, A. H. Morales, and J. Striker, On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope, arXiv preprint arXiv:1510.03357 [math.CO], 2015-2019.
• László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
• Index entries for linear recurrences with constant coefficients, signature (9,-28,35,-15,1).

Crossrefs

Cf. A000108, A001519, A011782, A024175, A033191, A080937, A080938, A124302.

Cf. square array in A080934.

Programs

Magma

m:=28; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5)));

Mathematica

CoefficientList[Series[(1 - 8 x + 21 x^2 - 20 x^3 + 5 x^4)/(1 - 9 x + 28 x^2 - 35 x^3 + 15 x^4 - x^5), {x, 0, 27}], x]

PARI

Vec((1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5)+O(x^28))

Formula

G.f.: (1-3*x+x^2)*(1-5*x+5*x^2)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

G.f.: 1/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x))))))))). - Philippe Deléham, Mar 14 2013

a(n) = A000108(n) + Sum_{k=1..n} (4*binomial(2*n, n+11*k) - binomial(2*n+2, n+11*k+1)). - Greg Dresden, Jan 28 2023