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A098334 Expansion of 1/sqrt(1-2x+17x^2).

%I A098334 #39 Aug 30 2024 02:55:36
%S A098334 1,1,-7,-23,49,401,41,-5767,-11423,65569,299353,-441847,-5511791,
%T A098334 -3665999,79937417,212712857,-861871423,-5076450239,3966949049,
%U A098334 89482678313,110424995569,-1233175514671,-4202194115863,11830822055353,91629438996001,-13485315511199
%N A098334 Expansion of 1/sqrt(1-2x+17x^2).
%C A098334 Central coefficients of (1+x-4x^2)^n.
%C A098334 Binomial transform of 1/sqrt(1+16x^2), or (1,0,-8,0,96,0,-1280,...).
%C A098334 Binomial transform is A098337.
%H A098334 Vincenzo Librandi, <a href="/A098334/b098334.txt">Table of n, a(n) for n = 0..200</a>
%H A098334 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.
%H A098334 Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%F A098334 E.g.f.: exp(x)*BesselI(0, 4*I*x), I=sqrt(-1);
%F A098334 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k, k)*(-4)^k;
%F A098334 a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*(-4)^k;
%F A098334 a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k, floor(k/2))*cos(Pi*k/2)*2^k.
%F A098334 D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +17*(n-1)*a(n-2)=0. - _R. J. Mathar_, Nov 24 2012
%F A098334 Lim sup n->oo |a(n)|^(1/n) = sqrt(17). - _Vaclav Kotesovec_, Feb 09 2014
%F A098334 a(n) = hypergeom([-n/2, 1/2-n/2], [1], -16). - _Peter Luschny_, Sep 18 2014
%F A098334 a(n) = (sqrt(17))^n*P(n,1/sqrt(17)), where P(n,x) is the Legendre polynomial of degree n. - _Peter Bala_, Mar 18 2018
%p A098334 a := n -> hypergeom([-n/2, 1/2-n/2], [1], -16);
%p A098334 seq(round(evalf(a(n),99)), n=0..28); # _Peter Luschny_, Sep 18 2014
%t A098334 CoefficientList[Series[1/Sqrt[1-2*x+17*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 09 2014 *)
%o A098334 (PARI) x='x+O('x^99); Vec(1/(1-2*x+17*x^2)^(1/2)) \\ _Altug Alkan_, Mar 18 2018
%Y A098334 Cf. A098331, A098332, A098333.
%K A098334 easy,sign
%O A098334 0,3
%A A098334 _Paul Barry_, Sep 03 2004