A086990 - OEIS

%I #26 Jul 21 2023 05:24:22

%S 0,1,-2,3,-4,6,-10,15,-20,30,-52,78,-96,144,-282,423,-420,630,-1660,

%T 2490,-1304,1956,-11332,16998,3896,-5844,-95240,142860,157160,-235740,

%U -983610,1475415,2634300,-3951450,-11751660,17627490,38381160,-57571740

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

%C Series reversion of Sum_{k>=0} a(k)x^k is x(Sum_{k>=0} A007051(k)x^k).

%C G.f. A(x) = Sum_{k>=0} a(k)x^k satisfies 0 = x - (4x+1)*A(x) + (3x+2)*A(x)^2.

%C If A(x)=g.f., then y=x/A(x)-2x satisfies x^2 = y^2 - y.

%H Vincenzo Librandi, <a href="/A086990/b086990.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1+4x-sqrt(1+4x^2))/(4+6x).

%F 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 From _Gary W. Adamson_, Jan 05 2012: (Start)

%F a(n) is the upper left term of (-1)*M^n, where M = an infinite square production matrix as follows:

%F   -1, -1, 0, 0, 0, 0, ...

%F   -1, 1, -1, 0, 0, 0, ...

%F   -1, 1, 1, -1, 0, 0, ...

%F   -1, 1, 1, 1, -1, 0, ...

%F   -1, 1, 1, 1, 1, -1, ...

%F   ... (End)

%F 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 Lim sup n->infinity |a(n)|^(1/n) = 2. - _Vaclav Kotesovec_, Feb 09 2014

%e a(5) = 6 = upper left term of (-1)*M^5. - _Gary W. Adamson_, Jan 05 2012

%t CoefficientList[Series[(1 + 4 x - Sqrt[1 + 4 x^2])/(4 + 6 x), {x, 0, 50}], x] (* _Harvey P. Dale_, Mar 24 2011 *)

%o (PARI) a(n)=polcoeff((1+4*x-sqrt(1+4*x^2+x*O(x^n)))/(4+6*x),n)

%K sign

%O 0,3

%A _Michael Somos_, Jul 27 2003