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%I #47 Feb 10 2022 06:15:14
%S 1,1,-1,-5,-5,11,41,29,-125,-365,-131,1409,3301,-155,-15625,-29485,
%T 16115,170035,254525,-309775,-1813055,-2064655,4617755,18909175,
%U 14903725,-61552739,-192390589,-81290561,767919595,1901796395,28588201
%N Expansion of 1/sqrt(1 - 2*x + 5*x^2).
%C Central coefficients of (1+x-x^2)^n. Binomial transform of 1/sqrt(1+4x^2), or (1,0,-2,0,6,0,-20,...). Binomial transform is A098335. (-1)^nA098331(n) is the inverse binomial transform of (1,0,-2,0,6,0,-20,...).
%C Hankel transform is 2^n*(-1)^C(n+1,2). Hankel transform of 0,1,1,-1,-5,-5,... is F(n)*(-1)^C(n+2,2)*(2^n+0^n)/2. - _Paul Barry_, Jan 13 2009
%H Robert Israel, <a href="/A098331/b098331.txt">Table of n, a(n) for n = 0..2865</a>
%H 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%F E.g.f.: exp(x)*BesselI(0, 2*i*x), i=sqrt(-1);
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*(-1)^k;
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*(-1)^k;
%F a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k, k/2)*cos(pi*k/2).
%F D-finite with recurrence: a(0)=a(1)=1, a(n) = ((2n-1)a(n-1)-5(n-1)a(n-2))/n. - _T. D. Noe_, Oct 19 2005
%F a(n) = hypergeom([-n/2, 1/2-n/2], [1], -4). - _Peter Luschny_, Sep 18 2014
%F a(n) ~ 5^(n/2 + 1/4) * cos((Pi*n - arctan(1/2) - n*arctan(4/3))/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 31 2017
%F a(n) = (sqrt(5))^n*P(n,1/sqrt(5)), where P(n,x) is the Legendre polynomial of degree n. Note the general result (sqrt(4*m+1))^n*P(n, 1/sqrt(4*m+1)) = Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)(-m)^k due to Catalan. - _Peter Bala_, Mar 18 2018
%F G.f.: 1/(1 - x + 2*x^2/(1 - x + x^2/(1 - x + x^2/(1 - x + x^2/(1 - ...))))), a continued fraction. - _Ilya Gutkovskiy_, Nov 19 2021
%F From _Peter Bala_, Feb 08 2022: (Start)
%F G.f.: A(x) = Sum_{n >= 0} (-1)^n*binomial(2*n,n)*x^(2*n)/(1 - x)^(2*n+1).
%F a(n)^2 = Sum_{k = 0..n} (-1)^k*5^(n-k)*binomial(2*k,k)*binomial(n,k)* binomial(n+k,k).
%F Sum_{n >= 0} (-1)^n*binomial(2*n,n)^2 * x^n/(1 - 5*x)^(2*n+1) = 1 + x + x^2 + 25*x^3 + 25*x^4 + 121*x^5 + ... is the g.f. of a(n)^2.
%F The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all prime p and positive integers n and k. (End)
%p A098331 := n -> hypergeom([-n/2, 1/2-n/2], [1], -4);
%p seq(round(evalf(A098331(n),99)),n=0..30); # _Peter Luschny_, Sep 18 2014
%p f:= gfun:-rectoproc({(5*n+5)*a(n)+(-3-2*n)*a(n+1)+(n+2)*a(n+2), a(0) = 1, a(1) = 1},a(n),remember):
%p map(f, [$0..50]); # _Robert Israel_, Jan 30 2018
%t a=b=1; Join[{a, b}, Table[c=((2n-1)b-5(n-1)a)/n; a=b; b=c; c, {n, 2, 30}]] (Noe)
%t CoefficientList[Series[1/Sqrt[1-2x+5x^2],{x,0,40}],x] (* _Harvey P. Dale_, Aug 17 2015 *)
%o (PARI) my(x='x+O('x^99)); Vec(1/(1-2*x+5*x^2)^(1/2)) \\ _Altug Alkan_, Mar 18 2018
%Y Cf. A098332, A098333, A098334.
%K easy,sign
%O 0,4
%A _Paul Barry_, Sep 03 2004
%E Corrected by _T. D. Noe_, Oct 19 2005
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