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%I A086456 #13 Jan 30 2020 21:29:15
%S A086456 1,-1,-2,-6,-22,-90,-394,-1806,-8558,-41586,-206098,-1037718,-5293446,
%T A086456 -27297738,-142078746,-745387038,-3937603038,-20927156706,
%U A086456 -111818026018,-600318853926,-3236724317174,-17518619320890,-95149655201962
%N A086456 Expansion of (1 + x + sqrt(1 - 6*x + x^2))/2 in powers of x.
%C A086456 Series reversion of x(Sum_{k>=0} a(k)x^k) is x(Sum_{k>=0} A003168(k)x^k).
%C A086456 G.f. A(x) = Sum_{k>=0} a(k)x^k satisfies 0 = 2*x - (x + 1)*A(x) + A(x)^2.
%H A086456 Foissy, Loic, <a href="https://doi.org/10.1007/s10801-012-0358-0">Algebraic structures on double and plane posets</a>, J. Algebr. Comb. 37, No. 1, 39-66 (2013).
%F A086456 G.f.: (1 + x + sqrt(1 - 6*x + x^2))/2. (= 1/g.f. A001003)
%F A086456 D-finite with recurrence: n*a(n) + 3*(-2*n + 3)*a(n-1) + (n-3)*a(n-2) = 0. - _R. J. Mathar_, Jul 23 2017
%t A086456 ReciprocalSeries[ser_, n_] := CoefficientList[ Series[1/ser, {x, 0, n}], x];
%t A086456 LittleSchroeder := (1 + x - Sqrt[1 - 6 x + x^2])/(4 x); (* A001003 *)
%t A086456 ReciprocalSeries[LittleSchroeder, 22] (* _Peter Luschny_, Jan 10 2019 *)
%o A086456 (PARI) a(n)=polcoeff((1+x+sqrt(1-6*x+x^2+x*O(x^n)))/2,n)
%Y A086456 A minor variation of A006318. a(n)=-A006318(n-1), n>0. a(n)=A085403(n), n>1.
%Y A086456 Cf. A001003.
%K A086456 sign
%O A086456 0,3
%A A086456 _Michael Somos_, Jul 20 2003

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