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A320825 Expansion of -(10*x^2 - 6*x + 1)*sqrt(1 - 4*x)/(3*x - 1)^2.

%I A320825 #24 Sep 08 2022 08:46:23
%S A320825 -1,2,1,0,-3,-10,-17,28,435,2710,13489,60392,254211,1028250,4046977,
%T A320825 15621932,59463209,224047866,838012755,3118339056,11563677321,
%U A320825 42790868982,158180470803,584617335420,2161733579313,8001589660746,29660171058675,110136151678696,409773163539325
%N A320825 Expansion of -(10*x^2 - 6*x + 1)*sqrt(1 - 4*x)/(3*x - 1)^2.
%H A320825 G. C. Greubel, <a href="/A320825/b320825.txt">Table of n, a(n) for n = 0..1000</a>
%F A320825 a(n) = c(n)*h(n) where c(n) = Catalan(n)*((n-3)*(n+1))/((2*n-3)*(2*n-1)) and h(n) = hypergeom([1, -n], [5/2 - n], 3/4).
%F A320825 A320825(n) = A320826(n) + A320827(n).
%F A320825 D-finite with recurrence: n*a(n) +(-13*n+15)*a(n-1) +4*(16*n-37)*a(n-2) +2*(-71*n+245)*a(n-3) +60*(2*n-9)*a(n-4)=0. - _R. J. Mathar_, Jan 23 2020
%p A320825 c := n -> catalan(n)*(n-3)*(n+1)/((2*n-3)*(2*n-1)):
%p A320825 h := n -> hypergeom([1, -n], [5/2 - n], 3/4): A320825 := n -> c(n)*h(n):
%p A320825 seq(simplify(A320825(n)), n=0..28);
%t A320825 CoefficientList[Series[-(10x^2 - 6x + 1) Sqrt[1 - 4x]/(3x - 1)^2, {x, 0, 28}], x]
%o A320825 (PARI) x='x+O('x^40); Vec(-(10*x^2-6*x+1)*sqrt(1-4*x)/(1-3*x)^2) \\ _G. C. Greubel_, Oct 27 2018
%o A320825 (Magma) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(-(10*x^2-6*x+1)*Sqrt(1-4*x)/(1-3*x)^2)); // _G. C. Greubel_, Oct 27 2018
%Y A320825 Cf. A126079, A320826, A320827.
%K A320825 sign
%O A320825 0,2
%A A320825 _Peter Luschny_, Oct 22 2018