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

1, -1, -2, -6, -22, -90, -394, -1806, -8558, -41586, -206098, -1037718, -5293446, -27297738, -142078746, -745387038, -3937603038, -20927156706, -111818026018, -600318853926, -3236724317174, -17518619320890, -95149655201962

Offset: 0

Author

Michael Somos, Jul 20 2003

Keywords

• sign

Comments

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

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

Links

• Foissy, Loic, Algebraic structures on double and plane posets, J. Algebr. Comb. 37, No. 1, 39-66 (2013).

Crossrefs

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

Cf. A001003.

Programs

Mathematica

ReciprocalSeries[ser_, n_] := CoefficientList[ Series[1/ser, {x, 0, n}], x];
LittleSchroeder := (1 + x - Sqrt[1 - 6 x + x^2])/(4 x); (* A001003 *)
ReciprocalSeries[LittleSchroeder, 22] (* Peter Luschny, Jan 10 2019 *)

PARI

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

Formula

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

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