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A115144 Fifth convolution of A115140.

%I A115144 #16 Mar 05 2023 11:01:44
%S A115144 1,-5,5,0,0,-1,-5,-20,-75,-275,-1001,-3640,-13260,-48450,-177650,
%T A115144 -653752,-2414425,-8947575,-33266625,-124062000,-463991880,
%U A115144 -1739969550,-6541168950,-24647883000,-93078189750,-352207870014,-1335293573130,-5071418015120,-19293438101000
%N A115144 Fifth convolution of A115140.
%H A115144 Seiichi Manyama, <a href="/A115144/b115144.txt">Table of n, a(n) for n = 0..1669</a>
%F A115144 O.g.f.: 1/c(x)^5 = P(6, x) - x*P(5, 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(6, x)=1-4*x+3*x^2 and P(5, x)=1-3*x+x^2.
%F A115144 a(n) = -C5(n-5), n>=5, with C5(n) = A000344(n+2) (fifth convolution of Catalan numbers). a(0)=1, a(1)=-5, a(2)=5, a(3)=0=a(4). [1, -5, 5] is row n=5 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
%F A115144 D-finite with recurrence +n*(n-5)*a(n) -2*(n-3)*(2*n-7)*a(n-1)=0. - _R. J. Mathar_, Sep 23 2021
%F A115144 From _Peter Bala_, Mar 05 2023: (Start)
%F A115144 a(n) = binomial(2*n - 6, n) - binomial(2*n - 6, n + 1).
%F A115144 a(n) = = -5/(n - 5)*binomial(2*n - 6, n) for n != 5.
%F A115144 a(n) = -A000344(n-3) for n >= 5. (End)
%t A115144 CoefficientList[Series[(1-5*x+5*x^2 +(1-3*x+x^2)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* _G. C. Greubel_, Feb 12 2019 *)
%o A115144 (PARI) my(x='x+O('x^30)); Vec((1-5*x+5*x^2 +(1-3*x+x^2)*sqrt(1-4*x))/2) \\ _G. C. Greubel_, Feb 12 2019
%o A115144 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-5*x+5*x^2 +(1-3*x+x^2)*Sqrt(1-4*x))/2 )); // _G. C. Greubel_, Feb 12 2019
%o A115144 (Sage) ((1-5*x+5*x^2 +(1-3*x+x^2)*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 12 2019
%Y A115144 Cf. A000344, A115139 - A115143, A115145 - A115149.
%K A115144 sign,easy
%O A115144 0,2
%A A115144 _Wolfdieter Lang_, Jan 13 2006