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 A086456 Expansion of (1 + x + sqrt(1 - 6*x + x^2))/2 in powers of x. 4

%I

%S 1,-1,-2,-6,-22,-90,-394,-1806,-8558,-41586,-206098,-1037718,-5293446,

%T -27297738,-142078746,-745387038,-3937603038,-20927156706,

%U -111818026018,-600318853926,-3236724317174,-17518619320890,-95149655201962

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

%C Series reversion of x(Sum_{k>=0} a(k)x^k) is x(Sum_{k>=0} A003168(k)x^k).

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

%H 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 G.f.: (1 + x + sqrt(1 - 6*x + x^2))/2. (= 1/g.f. A001003)

%F D-finite: n*a(n) + 3*(-2*n + 3)*a(n-1) + (n-3)*a(n-2) = 0. - _R. J. Mathar_, Jul 23 2017

%t ReciprocalSeries[ser_, n_] := CoefficientList[ Series[1/ser, {x, 0, n}], x];

%t LittleSchroeder := (1 + x - Sqrt[1 - 6 x + x^2])/(4 x); (* A001003 *)

%t ReciprocalSeries[LittleSchroeder, 22] (* _Peter Luschny_, Jan 10 2019 *)

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

%Y A minor variation of A006318. a(n)=-A006318(n-1), n>0. a(n)=A085403(n), n>1.

%Y Cf. A001003.

%K sign

%O 0,3

%A _Michael Somos_, Jul 20 2003

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Last modified December 8 12:07 EST 2019. Contains 329862 sequences. (Running on oeis4.)