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

-1, 2, 1, 0, -3, -10, -17, 28, 435, 2710, 13489, 60392, 254211, 1028250, 4046977, 15621932, 59463209, 224047866, 838012755, 3118339056, 11563677321, 42790868982, 158180470803, 584617335420, 2161733579313, 8001589660746, 29660171058675, 110136151678696, 409773163539325

Offset: 0

Author

Peter Luschny, Oct 22 2018

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A126079, A320826, A320827.

Programs

Magma

m:=40; R:=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

Maple

c := n -> catalan(n)*(n-3)*(n+1)/((2*n-3)*(2*n-1)):
h := n -> hypergeom([1, -n], [5/2 - n], 3/4): A320825 := n -> c(n)*h(n):
seq(simplify(A320825(n)), n=0..28);

Mathematica

CoefficientList[Series[-(10x^2 - 6x + 1) Sqrt[1 - 4x]/(3x - 1)^2, {x, 0, 28}], x]

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

Formula

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).

A320825(n) = A320826(n) + A320827(n).

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