A115143 a(n) = -4*binomial(2*n-5, n-4)/n for n > 0 and a(0) = 1.

1, -4, 2, 0, -1, -4, -14, -48, -165, -572, -2002, -7072, -25194, -90440, -326876, -1188640, -4345965, -15967980, -58929450, -218349120, -811985790, -3029594040, -11338026180, -42550029600, -160094486370, -603784920024, -2282138106804, -8643460269248, -32798844771700

Offset: 0

Author

Wolfdieter Lang, Jan 13 2006

Keywords

• sign
• easy

Comments

Previous name: Fourth convolution of A115140.

a(n+4) := - convolution ( A000108(n+1) ), n=0,1,... - Tilman Neumann, Jan 05 2009

Self-convolution of A115141. - R. J. Mathar, Sep 26 2012

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1669

Crossrefs

Cf. A115139 - A115142, A115144 - A115149, A099376, A000108.

Programs

Magma

[1,-4,2] cat [-4*Binomial(2*n-5,n-4)/n: n in [3..30]]; // G. C. Greubel, Feb 12 2019

Maple

A115143 := n -> `if`(n=0, 1, -4*binomial(2*n-5,n-4)/n):
seq(A115143(n), n=0..28); # Peter Luschny, Feb 27 2017
A115143List := proc(m) local A, P, n; A := [1,-4,2,0]; P := [-1,0];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A115143List(27); # Peter Luschny, Mar 26 2022

Mathematica

Join[{1},Table[-4*Binomial[2n-5,n-4]/n,{n,30}]] (* Harvey P. Dale, Dec 01 2017 *)
CoefficientList[Series[(1-4*x+2*x^2+(1-2*x)*Sqrt[1-4*x])/2, {x,0,30}], x] (* G. C. Greubel, Feb 12 2019 *)

PARI

my(x='x+O('x^30)); Vec((1-4*x+2*x^2 +(1-2*x)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

Sage

[1,-4,2] + [-4*binomial(2*n-5,n-4)/n for n in (3..30)] # G. C. Greubel, Feb 12 2019

Formula

O.g.f.: 1/c(x)^4 = P(5, x) - x*P(4, 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(5, x) = 1-3*x+x^2 and P(4, x) = 1-2*x.

a(n) = -C4(n-4), n>=4, with C4(n) := A002057(n) (fourth convolution of Catalan numbers). a(0)=1, a(1)=-4, a(2)=2, a(3)=0. [1, -4, 2] is row n=4 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.

E.g.f.: 1 - 3*x + 1/2*x^2 - x*Q(0), where Q(k)= 1 - 2*x/(k+2 - (k+2)*(2*k+1)/(2*k+1 - (k+2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 28 2013

D-finite with recurrence n*(n-4)*a(n) -2*(2*n-5)*(n-3)*a(n-1)=0. - R. J. Mathar, Sep 15 2024

Extensions

Simpler name from Peter Luschny, Feb 27 2017