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%I #11 Sep 08 2022 08:45:23
%S 1,-9,27,-30,9,0,0,0,0,-1,-9,-54,-273,-1260,-5508,-23256,-95931,
%T -389367,-1562275,-6216210,-24582285,-96768360,-379629720,-1485507600,
%U -5801732460,-22626756594,-88152205554,-343176898988,-1335293573130,-5193831553416
%N Ninth convolution of A115140.
%H Seiichi Manyama, <a href="/A115148/b115148.txt">Table of n, a(n) for n = 0..1671</a>
%F O.g.f.: 1/c(x)^9 = P(10, x) - x*P(9, 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(10, x)=1-8*x+21*x^2-20*x^3+5*x^4 and P(9, x)=1-7*x+15*x^2-10*x^3+x^4.
%F a(n) = -C9(n-9), n>=9, with C9(n) = A001392(n+4) (eighth convolution of Catalan numbers). a(0)=1, a(1)=-9, a(2)=27, a(3)=-30, a(4)=9, a(5)=a(6)=a(7)=a(8)=0. [1, -9, 27, -30, 9] is row n=9 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
%t CoefficientList[Series[(1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3 +x^4)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* _G. C. Greubel_, Feb 12 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2 -10*x^3+x^4)*sqrt(1-4*x))/2) \\ _G. C. Greubel_, Feb 12 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3+x^4)*Sqrt(1-4*x))/2 )); // _G. C. Greubel_, Feb 12 2019
%o (Sage) ((1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3+x^4) *sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 12 2019
%Y Cf. A115139 - A115147, A115149.
%K sign,easy
%O 0,2
%A _Wolfdieter Lang_, Jan 13 2006
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