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A115142 Third convolution of A115140.

%I A115142 #26 Sep 08 2022 08:45:23
%S A115142 1,-3,0,-1,-3,-9,-28,-90,-297,-1001,-3432,-11934,-41990,-149226,
%T A115142 -534888,-1931540,-7020405,-25662825,-94287120,-347993910,-1289624490,
%U A115142 -4796857230,-17902146600,-67016296620,-251577050010,-946844533674,-3572042254128,-13505406670700
%N A115142 Third convolution of A115140.
%H A115142 Seiichi Manyama, <a href="/A115142/b115142.txt">Table of n, a(n) for n = 0..1668</a>
%F A115142 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.
%F A115142 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.
%F A115142 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
%F A115142 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
%t A115142 CoefficientList[Series[(2*(1-2*x)-(1-x)*(1-Sqrt[1-4*x]))/2, {x, 0, 30}], x] (* _G. C. Greubel_, Feb 12 2019 *)
%o A115142 (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
%o A115142 (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
%o A115142 (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
%Y A115142 Cf. A000108, A000245, A071724.
%Y A115142 Cf. A115139, A115140, A115141, A115143 - A115149.
%K A115142 sign,easy
%O A115142 0,2
%A A115142 _Wolfdieter Lang_, Jan 13 2006