A131763 Series reversion of x*(1-4x)/(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

0,2

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

Links

Table of n, a(n) for n=0..20.

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.

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

Conjecture: (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)

Example

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

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 *)

Prog

(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 */

Crossrefs

Cf. A086810, A103211.

Sequence in context: A216171 A054879 A333090 * A006199 A083063 A012163

Adjacent sequences: A131760 A131761 A131762 * A131764 A131765 A131766

Keyword

nonn

Author

Philippe Deléham, Oct 29 2007, Nov 06 2007

Extensions

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

Status

approved