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%I #30 Sep 08 2022 08:45:23
%S 1,-7,14,-7,0,0,0,-1,-7,-35,-154,-637,-2548,-9996,-38760,-149226,
%T -572033,-2187185,-8351070,-31865925,-121580760,-463991880,
%U -1771605360,-6768687870,-25880277150,-99035193894,-379300783092,-1453986335186,-5578559816632,-21422369201800
%N Seventh convolution of A115140.
%H Seiichi Manyama, <a href="/A115146/b115146.txt">Table of n, a(n) for n = 0..1670</a>
%F O.g.f.: 1/c(x)^7 = P(8, x) - x*P(7, 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(8, x)=1-6*x+10*x^2-4*x^3 and P(7, x)=1-5*x+6*x^2-x^3.
%F a(n) = -C7(n-7), n>=7, with C7(n):=A000588(n+3) (seventh convolution of Catalan numbers). a(0)=1, a(1)=-7, a(2)=14, a(3)=-7, a(4)=a(5)=a(6)=0. [1, -7, 14, -7] is row n=7 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
%t CoefficientList[Series[(1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3) *Sqrt[1-4*x])/2, {x, 0, 30}], x] (* _G. C. Greubel_, Feb 12 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3) *sqrt(1-4*x))/2) \\ _G. C. Greubel_, Feb 12 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3)*Sqrt(1-4*x))/2 )); // _G. C. Greubel_, Feb 12 2019
%o (Sage) ((1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3)*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, A115145, A115147, A115148, A115149.
%K sign,easy
%O 0,2
%A _Wolfdieter Lang_, Jan 13 2006
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