A131763 Series reversion of x*(1-4*x)/(1-x) is x*A(x) where A(x) is the generating function.

1, 3, 21, 183, 1785, 18651, 204141, 2310447, 26819121, 317530227, 3819724293, 46553474919, 573608632233, 7133530172619, 89423593269213, 1128765846337887, 14334721079385441, 183021615646831587, 2347944226115977461, 30250309354902101271, 391241497991342192985

Offset: 0

Author

Philippe Deléham, Oct 29 2007, Nov 06 2007

Keywords

• nonn

Comments

The Hankel transform of this sequence is 12^C(n+1,2).

Number of Dyck n-paths with two colors of up (U,u) and two colors of down (D,d) avoiding UD. - David Scambler, Jun 24 2013

Number of small Schröder n-paths with 3 types of up steps (i.e., lattice paths from (0,0) to (2n,0) using steps U1=U2=U3=(1,1), F=(2,0), D=(1,-1), with no F steps on the x-axis). - Yu Hin Au, Dec 05 2019

Examples

			G.f. = 1 + 3*x + 21*x^2 + 183*x^3 + 1785*x^4 + 18651*x^5 + ... - _Michael Somos_, Jul 27 2022
		

Links

• J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, p_n(3).
• 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: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=3, b=4.

Crossrefs

Cf. A086810, A103211.

Cf. for series reversion of x*(1-r*x)/(1-x): A001003 (r=2), A107841 (r=3), this sequence (r=4), A131765 (r=5), A131846 (r=6), A131926 (r=7), A131869 (r=8), A131927 (r=9).

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[x*(1-4*x)/(1-x), {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Mar 30 2015 *)
Table[(-1)^n Hypergeometric2F1[-n, n + 1, 2, 4], {n, 0, 20}] (* Peter Luschny, Jan 08 2018 *)
a[ n_] := SeriesCoefficient[(1 + x - Sqrt[1 - 14*x + x^2])/(8*x), {x, 0, n}]; (* Michael Somos, Jul 27 2022 *)
a[ n_] := (-1)^n * Hypergeometric2F1[ -n, n+1, 2, 4]; (* Michael Somos, Mar 15 2024 *)

PARI

Vec(serreverse(x*(1-4*x)/(1-x)+ O(x^30))) \\ Michel Marcus, Mar 30 2015

PARI

{a(n) = if(n<0, 0, n++; polcoeff(serreverse(x*(1-4*x)/(1-x) + x*O(x^n)), n))}; /* Michael Somos, Jul 27 2022 */

PARI

{a(n) = if(n<0, -a(-1-n), polcoeff(2/(1 + x + sqrt(1 - 14*x + x^2 + x*O(x^n))), n))}; /* Michael Somos, Mar 15 2024 */

Formula

a(n) = Sum_{0<=k<=n} A086810(n,k)*3^k.

a(n) = (3/4)*A103211(n) for n>0.

a(n) = -a(n-1)+4*Sum_{i=0..n-1} a(i)*a(n-i-1), a(0)=1. - Vladimir Kruchinin, Mar 30 2015

D-finite with recurrence: (n+1)*a(n) +7*(-2*n+1)*a(n-1) +(n-2)*a(n-2)=0. - R. J. Mathar, Aug 16 2015

a(n) = (-1)^n*hypergeom([-n, n + 1], [2], 4). - Peter Luschny, Jan 08 2018

G.f.: (1 + x - sqrt(1 - 14*x + x^2))/(8*x). - Michael Somos, Jul 27 2022

From Michael Somos, Mar 15 2024: (Start)

Given g.f. A(x) and y = 2*x*A(-x^2), then y-1/y = (x-1/x)/2.

If a(n) := -a(-1-n) for n<0, then 0 = a(n)*(+a(n+1) -35*a(n+2) +4*a(n+3)) +a(n+1)*(+7*a(n+1) +194*a(n+2) -35*a(n+3)) +a(n+2)*(+7*a(n+2) +a(n+3)) for all n in Z. (End)

Extensions

a(17) corrected by Mark van Hoeij, Jul 01 2010