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%I #47 Sep 08 2022 08:45:24
%S 1,-2,-2,28,-74,-92,1324,-3656,-4826,70228,-197372,-267896,3921724,
%T -11126936,-15347432,225505648,-643622906,-897078476,13214495764,
%U -37869162392,-53170602284,784672445368,-2255295815192,-3183829452272,47051201187676,-135537088268792,-192142210448216
%N Expansion of 1/sqrt(1+4*x+16*x^2).
%C Fourth binomial transform is expansion of 1/sqrt(1-4*x+16*x^2), A012000.
%H G. C. Greubel, <a href="/A116091/b116091.txt">Table of n, a(n) for n = 0..1000</a>
%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%H Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%F E.g.f.: exp(-2*x)*Bessel_I(0, 2*sqrt(-3)*x).
%F a(n) = Sum_{k=0..n} C(n,k)^2*(-3)^k.
%F O.g.f.: P(-1/2,4*x) with the o.g.f. P(x,z):=1/sqrt(1-2*x*z+z^2) for the Legendre polynomials. _Wolfdieter Lang_, Mar 10 2011.
%F G.f. A(x) = 1/(2*T(0)-4*x-1) where T(k)= 1 + 3*x/(1 - x/T(k+1)); (continued fraction, 2-step ). - _Sergei N. Gladkovskii_, Aug 23 2012
%F D-finite with recurrence: a(n+2) = -(16*(n+1)*a(n))/(n+2) - (2*(2*n+3)*a(n+1))/(n+2) with a(0)=1, a(1)=-2. - _Alexander R. Povolotsky_, Aug 23 2012
%F a(n) = (-4)^n*hypergeom([-n, 1+n], [1], 1/4). - _Peter Luschny_, May 09 2016
%F From _Peter Bala_, Nov 30 2021: (Start)
%F a(n) = (-4)^n^P(n,1/2), where P(n,x) is the n-th Legendre polynomial.
%F a(n) = (4/3)*(16^n)*Sum_{k >= n} C(k,n)^2*(-1/3)^k.
%F a(n) = (-3)^n*hypergeom([-n, -n], [1], -1/3).
%F a(n) = (4/3)*(-16/3)^n*hypergeom([n+1, n+1], [1], -1/3).
%F a(n) = [x^n] ((1 + x)*(3 - x))^n. (End)
%p a := n -> (-4)^n*hypergeom([-n,1+n],[1],1/4);
%p seq(simplify(a(n)), n=0..26); # _Peter Luschny_, May 09 2016
%t Table[4^n*LegendreP[n,-1/2],{n,0,30}] (* _Vaclav Kotesovec_, Jul 23 2013 *)
%t CoefficientList[Series[1/Sqrt[1+4x+16x^2],{x,0,30}],x] (* _Harvey P. Dale_, Jun 08 2015 *)
%o (PARI) Vec(1/sqrt(1+4*x+16*x^2+O(x^30))) \\ _M. F. Hasler_, Aug 25 2012
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1+4*x+16*x^2) )); // _G. C. Greubel_, May 09 2019
%o (Sage) (1/sqrt(1+4*x+16*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 09 2019
%Y Cf. A012000, A098332, A126869.
%K sign,easy
%O 0,2
%A _Paul Barry_, Feb 04 2006
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