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

%I #29 Sep 08 2022 08:46:23

%S -1,1,1,3,11,41,151,549,1977,7075,25229,89831,319881,1140523,4075321,

%T 14603243,52501659,189440937,686181711,2495243373,9109701699,

%U 33388293177,122840931891,453622854873,1681057537359,6250742452125,23316503569983,87236431248445

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

%H G. C. Greubel, <a href="/A320827/b320827.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 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.

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

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

%p ogf := x -> -sqrt(1 - 4*x)*(2*x - 1)/(3*x - 1);

%p ser := series(ogf(x), x, 30); seq(coeff(ser, x, k), k=0..27);

%p # By recurrence:

%p a := proc(n) option remember; if n <= 4 then return [-1,1,1,3,11][n+1] fi;

%p ((-90+66*n-12*n^2)*a(n-2)+(30-34*n+7*n^2)*a(n-1))/((n-4)*n) end:

%p seq(a(n), n=0..27);

%t 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}]

%t CoefficientList[Series[Sqrt[1-4*x]*(1-2*x)/(3*x-1), {x, 0, 40}], x] (* _G. C. Greubel_, Oct 27 2018 *)

%o (PARI) x='x+O('x^40); Vec(sqrt(1-4*x)*(1-2*x)/(3*x-1)) \\ _G. C. Greubel_, Oct 27 2018

%o (Magma) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt(1-4*x)*(1-2*x)/(3*x-1))); // _G. C. Greubel_, Oct 27 2018

%Y Cf. A002421, A029759, A320825, A320826.

%K sign

%O 0,4

%A _Peter Luschny_, Oct 23 2018

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Last modified July 16 05:19 EDT 2024. Contains 374343 sequences. (Running on oeis4.)