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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). 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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, AH Morales, J Striker, , On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope, arXiv preprint arXiv:1510.03357, 2015

Index entries for linear recurrences with constant coefficients, signature (9,-28,35,-15,1).

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

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]

PROG

(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))

(MAGMA) m:=28; R<x>:=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)));

CROSSREFS

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

Cf. square array in A080934.

Sequence in context: A287972 A243838 A242450 * A261592 A291824 A287973

Adjacent sequences:  A211213 A211214 A211215 * A211217 A211218 A211219

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, May 11 2012

STATUS

approved

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Last modified December 12 20:04 EST 2017. Contains 295954 sequences.