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A098333 Expansion of 1/sqrt(1 - 2x + 13x^2).

%I A098333 #41 Jan 30 2020 21:29:15
%S A098333 1,1,-5,-17,19,211,181,-2015,-5837,12259,91585,29965,-1033955,
%T A098333 -2347955,7953115,43864543,-11941037,-559875245,-942036911,5060812717,
%U A098333 21502740649,-20676139991,-307241918945,-344022187613
%N A098333 Expansion of 1/sqrt(1 - 2x + 13x^2).
%C A098333 Central coefficients of (1 + x - 3x^2)^n.
%C A098333 Binomial transform of 1/sqrt(1+12x^2), or (1,0,-6,0,54,0,-540,...).
%C A098333 Binomial transform is A012000.
%H A098333 Vincenzo Librandi, <a href="/A098333/b098333.txt">Table of n, a(n) for n = 0..200</a>
%H A098333 Hacène Belbachir, Abdelghani Mehdaoui, 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 A098333 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 A098333 E.g.f.: exp(x)*BesselI(0, 2*sqrt(-3)x);
%F A098333 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)(-3)^k;
%F A098333 a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)(-3)^k.
%F A098333 D-finite with recurrence: n*a(n) + (-2*n+1)*a(n-1) + 13*(n-1)*a(n-2) = 0. - _R. J. Mathar_, Nov 24 2012
%F A098333 Lim sup n->infinity |a(n)|^(1/n) = sqrt(13). - _Vaclav Kotesovec_, Feb 09 2014
%F A098333 a(n) = (sqrt(13))^n*P(n,1/sqrt(13)), where P(n,x) is the Legendre polynomial of degree n. - _Peter Bala_, Mar 18 2018
%F A098333 a(n) = hypergeom([1/2 - n/2, -n/2], [1], -12). - _Peter Luschny_, Mar 19 2018
%p A098333 a := n -> hypergeom([1/2 - n/2, -n/2], [1], -12):
%p A098333 seq(simplify(a(n)), n=0..23); # _Peter Luschny_, Mar 19 2018
%t A098333 CoefficientList[Series[1/Sqrt[1-2*x+13*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 09 2014 *)
%o A098333 (PARI) x='x+O('x^99); Vec(1/(1-2*x+13*x^2)^(1/2)) \\ _Altug Alkan_, Mar 18 2018
%Y A098333 Cf. A098331, A098332, A098334.
%K A098333 easy,sign
%O A098333 0,3
%A A098333 _Paul Barry_, Sep 03 2004