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%I #22 Sep 08 2022 08:45:24
%S 1,8,72,656,5992,54768,500688,4577568,41851560,382641200,3498428272,
%T 31985610720,292439802256,2673735097184,24445577182368,
%U 223502416896576,2043450657688872,18682977401318064,170815793235313968
%N Expansion of 1/(4*sqrt(1-4*x) - 3).
%C The g.f. is A(x)^2/(2*A(x)-A(x)^2) where A(x) is the g.f. of A076035.
%C The Hankel transform of this sequence is 8^n = [1, 8, 64, 512, 4096, ...]; the Hankel transform of the aerated sequence with g.f. 1/(1-8*x^2*c(x^2)) is also 8^n. - _Philippe Deléham_, Feb 13 2007
%H Vincenzo Librandi, <a href="/A115970/b115970.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: 1/(1-8*x*c(x)), where c(x) is the g.f. of A000108.
%F a(n) = Sum_{k=0..n} A106566(n, k)*8^k.
%F From _Philippe Deléham_, Feb 13 2007: (Start)
%F a(n) = (64*a(n-1) - 8*A000108(n-1))/7.
%F a(n) = Sum_{k=0..n} A039599(n,k)*7^k.
%F a(n) = Sum_{k=0..n} A106566(n,k)*8^k. (End)
%F D-finite with recurrence: 7*n*a(n) = 2*(46*n-21)*a(n-1) - 128*(2*n-3)*a(n-2). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) ~ 3*2^(6*n+1)/7^(n+1). - _Vaclav Kotesovec_, Oct 19 2012
%t CoefficientList[Series[1/(4*Sqrt[1-4*x]-3), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%o (PARI) my(x='x+O('x^20)); Vec(1/(4*sqrt(1-4*x)-3)) \\ _G. C. Greubel_, May 05 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 1/(4*Sqrt(1-4*x)-3) )); // _G. C. Greubel_, May 05 2019
%o (Sage) (1/(4*sqrt(1-4*x)-3)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, May 05 2019
%K easy,nonn
%O 0,2
%A _Paul Barry_, Feb 03 2006
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