A320827 G.f.: -sqrt(1 - 4*x)*(2*x - 1)/(3*x - 1).

-1, 1, 1, 3, 11, 41, 151, 549, 1977, 7075, 25229, 89831, 319881, 1140523, 4075321, 14603243, 52501659, 189440937, 686181711, 2495243373, 9109701699, 33388293177, 122840931891, 453622854873, 1681057537359, 6250742452125, 23316503569983, 87236431248445

Offset: 0

Author

Peter Luschny, Oct 23 2018

Keywords

• sign

Links

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

Crossrefs

Cf. A002421, A029759, A320825, A320826.

Programs

Magma

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt(1-4*x)*(1-2*x)/(3*x-1))); // G. C. Greubel, Oct 27 2018

Maple

ogf := x -> -sqrt(1 - 4*x)*(2*x - 1)/(3*x - 1);
ser := series(ogf(x), x, 30); seq(coeff(ser, x, k), k=0..27);
# By recurrence:
a := proc(n) option remember; if n <= 4 then return [-1,1,1,3,11][n+1] fi;
((-90+66*n-12*n^2)*a(n-2)+(30-34*n+7*n^2)*a(n-1))/((n-4)*n) end:
seq(a(n), n=0..27);

Mathematica

a[n_] := (-4)^n Binomial[3/2,n]((4/3)n - 2 + Hypergeometric2F1[1,-n, 5/2 - n, 3/4]); Table[a[n], {n, 0, 27}]
CoefficientList[Series[Sqrt[1-4*x]*(1-2*x)/(3*x-1), {x, 0, 40}], x] (* G. C. Greubel, Oct 27 2018 *)

PARI

x='x+O('x^40); Vec(sqrt(1-4*x)*(1-2*x)/(3*x-1)) \\ G. C. Greubel, Oct 27 2018

Formula

a(n) = (-4)^n*binomial(3/2, n)*((4/3)*n - 2 + hypergeom([1, -n], [5/2 - n], 3/4)).

D-finite with recurrence: a(n) = ((-90+66*n-12*n^2)*a(n-2) + (30-34*n+7*n^2)*a(n-1))/((n-4)*n) for n >= 5.

Expansion of -1/g.f. gives A029759.

a(n) = A320825(n) - A320826(n).