|
%I #73 Feb 21 2020 06:38:16
%S 1,2,10,62,430,3194,24850,199910,1649350,13879538,118669210,
%T 1027945934,9002083870,79568077034,708911026210,6359857112438,
%U 57403123415350,520895417047010,4749381474135850,43489017531266654,399755692955359630,3687437532852484442,34121911117572911410
%N Series reversion of x(1-3x)/(1-x).
%C 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}.
%C The Hankel transform of this sequence is 6^C(n+1,2). - _Philippe Deléham_, Oct 29 2007
%C 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
%C 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
%C 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
%C 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
%C 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
%H Fung Lam, <a href="/A107841/b107841.txt">Table of n, a(n) for n = 0..1000</a> [The first 200 terms were computed by Vincenzo Librandi]
%H J. Abate, W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Whitt/whitt2.html">Integer Sequences from Queueing Theory </a>, J. Int. Seq. 13 (2010), 10.5.5, p_n(2).
%H Yu Hin (Gary) Au, <a href="https://arxiv.org/abs/1912.00555">Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers</a>, arXiv:1912.00555 [math.CO], 2019.
%H Paul Barry, <a href="http://arxiv.org/abs/1312.0583">Embedding structures associated with Riordan arrays and moment matrices</a>, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
%H Z. Chen, H. Pan, <a href="http://arxiv.org/abs/1608.02448">Identities involving weighted Catalan-Schroder and Motzkin Paths</a>, arXiv:1608.02448 [math.CO], 2016; eq. (1.13), a=2, b=3.
%H Samuele Giraudo, <a href="http://arxiv.org/abs/1504.04529">Operads from posets and Koszul duality</a>, arXiv preprint arXiv:1504.04529 [math.CO], 2015.
%H F. Lam, <a href="http://arxiv.org/abs/1311.6395">Integral Invariance and Non-linearity Reduction for Proliferating Vorticity Scales in Fluid Dynamics</a>, arXiv:1311.6395 [physics.flu-dyn], 2013-2014.
%H F. Lam, <a href="http://arxiv.org/abs/1505.07723">Vorticity evolution in a rigid pipe of circular cross-section</a>, arXiv preprint arXiv:1505.07723 [physics.flu-dyn], 2015-2019.
%F G.f.: (1+x-sqrt(1-10x+x^2))/(6x).
%F a(n) = (1/(n+1))sum{k=0..n, C(n+1, k)C(2n-k, n)(-1)^k*3^(n-k)}.
%F a(n) = (1/(n+1))sum{k=0..n, C(n+1, k+1)C(n+k, k)(-1)^(n-k)*3^k}.
%F a(n) = sum{k=0..n, (1/(k+1))*C(n, k)C(n+k, k)(-1)^(n-k)*3^k}.
%F a(n) = sum{k=0..n, A088617(n, k)*(-1)^(n-k)*3^k}.
%F a(n) = Sum_{k>=0} A086810(n, k)*2^k. - _Philippe Deléham_, May 26 2005
%F a(n) = (2/3)*A103210(n) for n>0. - _Philippe Deléham_, Oct 29 2007
%F 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
%F From _Paul Barry_, May 15 2009: (Start)
%F G.f.: 1/(1-2x/(1-x-2x/(1-x-2x/(1-x-2x/(1-x-2x/(1-... (continued fraction).
%F G.f.: 1/(1-2x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-... (continued fraction). (End)
%F 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
%F 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
%F a(n) ~ sqrt(12+5*sqrt(6))*(5+2*sqrt(6))^n/(6*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 17 2012
%F 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
%t CoefficientList[Series[(1+x-Sqrt[1-10*x+x^2])/(6*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%o (PARI) x='x+O('x^66); Vec(serreverse(x*(1-3*x)/(1-x))) \\ _Joerg Arndt_, May 15 2013
%Y Cf. A001003.
%K easy,nonn
%O 0,2
%A _Paul Barry_, May 24 2005
|
