A107841 Series reversion of x*(1-3*x)/(1-x).

1, 2, 10, 62, 430, 3194, 24850, 199910, 1649350, 13879538, 118669210, 1027945934, 9002083870, 79568077034, 708911026210, 6359857112438, 57403123415350, 520895417047010, 4749381474135850, 43489017531266654, 399755692955359630, 3687437532852484442, 34121911117572911410

Offset: 0

Author

Paul Barry, May 24 2005

Keywords

• easy
• nonn

Comments

In general, the series reversion of x(1-r*x)/(1-x) has g.f. (1+x-sqrt(1+2*(1-2*r)*x+x^2))/(2*r) and general term given by a(n)=(1/(n+1))sum{k=0..n, C(n+1,k)C(2n-k,n)(-1)^k*r^(n-k)}; a(n)=(1/(n+1))sum{k=0..n, C(n+1,k+1)C(n+k,k)(-1)^(n-k)*r^k}; a(n)=sum{k=0..n, (1/(k+1))*C(n,k)C(n+k,k)(-1)^(n-k)*r^k}; a(n)=sum{k=0..n, A088617(n,k)*(-1)^(n-k)*r^k}.

The Hankel transform of this sequence is 6^C(n+1,2). - Philippe Deléham, Oct 29 2007

Number of Dyck n-paths with three colors of up (U,a,b) and one color of down (D) avoiding UD. - David Scambler, Jun 24 2013

This sequence is implied in the turbulence solutions of the incompressible Navier-Stokes equations in R^3. a(n) = numbers of realizable vorticity eddies in terms of initial conditions. - Fung Lam, Dec 31 2013

Conjugate sequence to this series is defined by series reversion of x(1+3*x)/(1+x), G.f.: ((x-1)-sqrt(1-10*x+ x^2))/(6*x). Conjugate sequence is the negation of this series except a(0). - Fung Lam, Jan 16 2014

Complete Chebyshev transform is G.f. = 3*F((1-x^2)/(1+x^2)), where F(x) is the g.f. of A107841. Real part of G.f. (= (1 - sqrt(3*x^4-2))/((1+x^2))) generates periodic sequence A056594. In general, for reversion of x*(1-r*x)/(1-x), r>=2, Real part of r*F((1-x^2)/(1+x^2)) (= (1 - sqrt(r*x^4 - r + 1))/(1+x^2)) generates A056594. - Fung Lam, Apr 29 2014

a(n) is the number of small Schröder n-paths with 2 types of up steps (i.e., lattice paths from (0,0) to (2n,0) using steps U1=U2=(1,1), F=(2,0), D=(1,-1), with no F steps on the x-axis). - Yu Hin Au, Dec 07 2019

Links

• Fung Lam, Table of n, a(n) for n = 0..1000 [The first 200 terms were computed by Vincenzo Librandi]
• J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, p_n(2).
• Yu Hin (Gary) Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
• Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
• Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016; eq. (1.13), a=2, b=3.
• Samuele Giraudo, Operads from posets and Koszul duality, arXiv preprint arXiv:1504.04529 [math.CO], 2015.
• F. Lam, Integral Invariance and Non-linearity Reduction for Proliferating Vorticity Scales in Fluid Dynamics, arXiv:1311.6395 [physics.flu-dyn], 2013-2014.
• F. Lam, Vorticity evolution in a rigid pipe of circular cross-section, arXiv preprint arXiv:1505.07723 [physics.flu-dyn], 2015-2019.

Crossrefs

Cf. A001003 (r=2), this sequence (r=3), A131763 (r=4), A131765 (r=5), A131846 (r=6), A131926 (r=7), A131869 (r=8), A131927 (r=9).

Programs

Maple

seq(simplify((-1)^n*hypergeom([-n, n + 1], [2], 3)), n=0..10); # Georg Fischer, Sep 14 2024 (from Peter Luschny's formula in A131763, with last parameter r=3)

Mathematica

CoefficientList[Series[(1+x-Sqrt[1-10*x+x^2])/(6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

PARI

x='x+O('x^66); Vec(serreverse(x*(1-3*x)/(1-x))) \\ Joerg Arndt, May 15 2013

Formula

G.f.: (1+x-sqrt(1-10x+x^2))/(6x).

a(n) = (1/(n+1))sum{k=0..n, C(n+1, k)C(2n-k, n)(-1)^k*3^(n-k)}.

a(n) = (1/(n+1))sum{k=0..n, C(n+1, k+1)C(n+k, k)(-1)^(n-k)*3^k}.

a(n) = sum{k=0..n, (1/(k+1))*C(n, k)C(n+k, k)(-1)^(n-k)*3^k}.

a(n) = sum{k=0..n, A088617(n, k)*(-1)^(n-k)*3^k}.

a(n) = Sum_{k>=0} A086810(n, k)*2^k. - Philippe Deléham, May 26 2005

a(n) = (2/3)*A103210(n) for n>0. - Philippe Deléham, Oct 29 2007

G.f.: 1/(1-2x/(1-3x/(1-2x/(1-3x/(1-2x/(1-3x/(1-2x/(1-3x........ (continued fraction). - Paul Barry, Dec 15 2008

From Paul Barry, May 15 2009: (Start)

G.f.: 1/(1-2x/(1-x-2x/(1-x-2x/(1-x-2x/(1-x-2x/(1-... (continued fraction).

G.f.: 1/(1-2x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-... (continued fraction). (End)

G.f.: 1/(1+x-3x/(1+x-3x/(1+x-3x/(1+x-3x/(1+x-3x/(1+... (continued fraction). - Paul Barry, Mar 18 2011

D-finite with recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ sqrt(12+5*sqrt(6))*(5+2*sqrt(6))^n/(6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

a(n+1) is the coefficient of x^(n+1) in 2*sum{j,1,n}((sum{k,1,n}a(k)x^k)^(j+1)), a(1)=1 with offset by 1. - Fung Lam, Dec 31 2013

The series reversion of x*(1 - r*x)/(1 - x) is D-finite with the general recurrence n*a(n) - (2*r-1)*(2*n-3)*a(n-1) + (n-3)*a(n-2) = 0 and with initial values a(1) = 1, a(2) = r-1, a(3) = (2*r-1)*(r-1). This sequence uses r=3, cf. crossrefs. - Georg Fischer, Sep 14 2024