A115142 Third convolution of A115140.

1, -3, 0, -1, -3, -9, -28, -90, -297, -1001, -3432, -11934, -41990, -149226, -534888, -1931540, -7020405, -25662825, -94287120, -347993910, -1289624490, -4796857230, -17902146600, -67016296620, -251577050010, -946844533674, -3572042254128, -13505406670700

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1668

Formula

O.g.f.: 1/c(x)^3 = P(4, x) - x*P(3, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(4, x)=1-2*x and P(3, x)=1-x.

a(n) = -C3(n-3), n >= 3, with C3(n):= A000245(n+1) (third convolution of Catalan numbers). a(0)=1, a(1)=-3, a(2)=0. [1, -3] is the row n=3 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.

D-finite with recurrence +n*(n-3)*a(n) -2*(n-2)*(2*n-5)*a(n-1)=0. - R. J. Mathar, Sep 23 2021

O.g.f.: (1/8)*(1 + sqrt(1 - 4*x))^3 = ( hypergeom([-1/4, -3/4], [-1/2], 4*x) )^2. - Peter Bala, Mar 04 2022

Mathematica

CoefficientList[Series[(2*(1-2*x)-(1-x)*(1-Sqrt[1-4*x]))/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((2*(1-2*x)-(1-x)*(1-sqrt(1-4*x)))/2) \\ G. C. Greubel, Feb 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2*(1-2*x)-(1-x)*(1-Sqrt(1-4*x)))/2 )); // G. C. Greubel, Feb 12 2019
(Sage) ((2*(1-2*x)-(1-x)*(1-sqrt(1-4*x)))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

Crossrefs

Cf. A000108, A000245, A071724.

Cf. A115139, A115140, A115141, A115143 - A115149.

Sequence in context: A325673 A083857 A353077 * A346203 A327126 A273083

Adjacent sequences: A115139 A115140 A115141 * A115143 A115144 A115145

Keyword

sign,easy

Author

Wolfdieter Lang, Jan 13 2006

Status

approved