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%I #17 Sep 08 2022 08:45:23
%S 1,-6,9,-2,0,0,-1,-6,-27,-110,-429,-1638,-6188,-23256,-87210,-326876,
%T -1225785,-4601610,-17298645,-65132550,-245642760,-927983760,
%U -3511574910,-13309856820,-50528160150,-192113383644,-731508653106,-2789279908316,-10649977831752,-40715807302800
%N Sixth convolution of A115140.
%H Seiichi Manyama, <a href="/A115145/b115145.txt">Table of n, a(n) for n = 0..1669</a>
%F O.g.f.: 1/c(x)^6 = P(7, x) - x*P(6, 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(7, x)=1-5*x+6*x^2-x^3 and P(6, x) = 1-4*x+3*x^2.
%F a(n) = -C6(n-6), n>=6, with C6(n) = A003517(n+2) (sixth convolution of Catalan numbers). a(0)=1, a(1)=-6, a(2)=9, a(3)=-2, a(4)=0=a(5). [1, -6, 9, -2] is row n=6 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
%F D-finite with recurrence +n*(n-6)*a(n) -2*(2*n-7)*(n-4)*a(n-1)=0. - _R. J. Mathar_, Sep 23 2021
%t CoefficientList[Series[(1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* _G. C. Greubel_, Feb 12 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2) *sqrt(1-4*x))/2) \\ _G. C. Greubel_, Feb 12 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt(1-4*x))/2 )); // _G. C. Greubel_, Feb 12 2019
%o (Sage) ((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 12 2019
%Y Cf. A115139, A115140, A115141, A115142, A115143.
%Y Cf. A115144, A115146, A115147, A115148, A115149.
%K sign,easy
%O 0,2
%A _Wolfdieter Lang_, Jan 13 2006
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