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A086990 Expansion of (1+4x-sqrt(1+4x^2))/(4+6x) in powers of x.

%I A086990 #26 Jul 21 2023 05:24:22
%S A086990 0,1,-2,3,-4,6,-10,15,-20,30,-52,78,-96,144,-282,423,-420,630,-1660,
%T A086990 2490,-1304,1956,-11332,16998,3896,-5844,-95240,142860,157160,-235740,
%U A086990 -983610,1475415,2634300,-3951450,-11751660,17627490,38381160,-57571740
%N A086990 Expansion of (1+4x-sqrt(1+4x^2))/(4+6x) in powers of x.
%C A086990 Series reversion of Sum_{k>=0} a(k)x^k is x(Sum_{k>=0} A007051(k)x^k).
%C A086990 G.f. A(x) = Sum_{k>=0} a(k)x^k satisfies 0 = x - (4x+1)*A(x) + (3x+2)*A(x)^2.
%C A086990 If A(x)=g.f., then y=x/A(x)-2x satisfies x^2 = y^2 - y.
%H A086990 Vincenzo Librandi, <a href="/A086990/b086990.txt">Table of n, a(n) for n = 0..1000</a>
%F A086990 G.f.: (1+4x-sqrt(1+4x^2))/(4+6x).
%F A086990 G.f.: (x-x^2*c(-x^2))/(1+x-x^2*c(-x^2)), c(x) the g.f. of A000108. - _Paul Barry_, Jun 17 2005
%F A086990 From _Gary W. Adamson_, Jan 05 2012: (Start)
%F A086990 a(n) is the upper left term of (-1)*M^n, where M = an infinite square production matrix as follows:
%F A086990   -1, -1, 0, 0, 0, 0, ...
%F A086990   -1, 1, -1, 0, 0, 0, ...
%F A086990   -1, 1, 1, -1, 0, 0, ...
%F A086990   -1, 1, 1, 1, -1, 0, ...
%F A086990   -1, 1, 1, 1, 1, -1, ...
%F A086990   ... (End)
%F A086990 D-finite with recurrence 2*n*a(n) +3*n*a(n-1) +8*(n-3)*a(n-2) +12*(n-3)*a(n-3)=0. - _R. J. Mathar_, Nov 24 2012
%F A086990 Lim sup n->infinity |a(n)|^(1/n) = 2. - _Vaclav Kotesovec_, Feb 09 2014
%e A086990 a(5) = 6 = upper left term of (-1)*M^5. - _Gary W. Adamson_, Jan 05 2012
%t A086990 CoefficientList[Series[(1 + 4 x - Sqrt[1 + 4 x^2])/(4 + 6 x), {x, 0, 50}], x] (* _Harvey P. Dale_, Mar 24 2011 *)
%o A086990 (PARI) a(n)=polcoeff((1+4*x-sqrt(1+4*x^2+x*O(x^n)))/(4+6*x),n)
%K A086990 sign
%O A086990 0,3
%A A086990 _Michael Somos_, Jul 27 2003