A210628 - OEIS

%I #57 Sep 08 2022 08:46:01

%S 1,-1,0,-1,0,-2,0,-5,0,-14,0,-42,0,-132,0,-429,0,-1430,0,-4862,0,

%T -16796,0,-58786,0,-208012,0,-742900,0,-2674440,0,-9694845,0,

%U -35357670,0,-129644790,0,-477638700,0,-1767263190,0,-6564120420,0,-24466267020,0

%N Expansion of (-1 + 2*x + sqrt( 1 - 4*x^2)) / (2*x) in powers of x.

%C Except for the leading term, the sequence is equal to -A097331(n). - _Fung Lam_, Mar 22 2014

%H Vincenzo Librandi, <a href="/A210628/b210628.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1 - (2*x) / (1 + sqrt( 1 - 4*x^2)) = 1 - (1 - sqrt( 1 - 4*x^2)) / (2*x).

%F G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = x*y^2 - (1 - 2*x) * (1 - y).

%F G.f. A(x) satisfies A( x / (1 + x^2) ) = 1 - x.

%F G.f. A(x) = 1 - x - x * (1 - A(x))^2 = 1 - 1/x + 1 / (1 - A(x)).

%F G.f. A(x) = 1 / (1 + x / (1 - 2*x + x * A(x))).

%F G.f. A(x) = 1 / (1 + x / (1 - x / (1 - x / (1 + x * A(x))))).

%F G.f. A(x) = 1 / (1 + x * A001405(x)). A126930(x) = 1 / (1 + x * A(x)).

%F G.f. A(x) = 1 - x / (1 - x^2 / (1 - x^2 / (1 - x^2 / ...))). - _Michael Somos_, Jan 02 2013

%F a(2*n) = 0 unless n=0, a(2*n + 1) = -A000108(n). a(n) = (-1)^n * A097331(n). a(n-1) = (-1)^floor(n/2) * A090192(n).

%F Convolution inverse of A210736. - _Michael Somos_, Jan 02 2013

%F G.f.: 2/( G(0) + 1), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1+2*x) - 2*x*(1+2*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1+2*x)*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 24 2013

%F D-finite with recurrence: (n+3)*a(n+2) = 4*n*a(n), a(0)=1, a(1)=-1. - _Fung Lam_, Mar 17 2014

%F For nonzero odd-power terms, a(n) = -2^(n+1)/(n+1)^(3/2)/sqrt(2*Pi)*(1+3/(4*n) + O(1/n^2)). (with contribution of Vaclav Kotesovec) - _Fung Lam_, Mar 17 2014

%e G.f. = 1 - x - x^3 - 2*x^5 - 5*x^7 - 14*x^9 - 42*x^11 - 132*x^13 - 429*x^15 + ...

%t CoefficientList[Series[1 - 2 x/(1 + Sqrt[1 - 4 x^2]), {x, 0, 45}], x] (* _Bruno Berselli_, Mar 25 2012 *)

%t a[ n_] := SeriesCoefficient[ (-1 + 2 x + Sqrt[1 - 4 x^2]) / (2 x), {x, 0, n}];

%o (PARI) {a(n) = polcoeff( (-1 + 2*x + sqrt( 1 - 4*x^2 + x^2 * O(x^n))) / (2*x), n)};

%o (PARI) {a(n) = if( n<1, n==0, polcoeff( serreverse( -x / (1 + x^2) + x * O(x^n)), n))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = 1 - x - x * (1 - A)^2); polcoeff( A, n))};

%o (Maxima) makelist(coeff(taylor(1-2*x/(1+sqrt(1-4*x^2)), x, 0, n), x, n), n, 0, 45); \\ _Bruno Berselli_, Mar 25 2012

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((-1 + 2*x + Sqrt(1-4*x^2))/(2*x))); // _G. C. Greubel_, Aug 11 2018

%Y Cf. A000108, A001405, A090192, A097331, A126930, A210736.

%K sign

%O 0,6

%A _Michael Somos_, Mar 25 2012