A091565 Expansion of (1 - x - sqrt(1 - 2*x + x^2 - 8*x^3)) / (4*x^2) in powers of x.

0, 1, 1, 1, 3, 7, 13, 29, 71, 163, 377, 913, 2219, 5375, 13189, 32677, 81167, 202523, 508273, 1280537, 3236275, 8207543, 20880893, 53263405, 136205527, 349137811, 896881641, 2308523809, 5953138875, 15378562415, 39791453685, 103115768885

Offset

0,5

Comments

Series reversion of g.f. A(x) is -A(-x).

Links

Table of n, a(n) for n=0..31.

Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.

Formula

G.f.: (1 - x - sqrt(1 - 2*x + x^2 - 8*x^3)) / (4*x^2) = 2*x / (1 - x + sqrt(1 - 2*x + x^2 - 8*x^3)) .

G.f. A(x) = y satisfies 2*(x*y)^2 + (x - 1)*y + x = 0.

a(n) = a(n-1) + 2*(a(1)*a(n-3) + a(2)*a(n-4) + ... + a(n-3)*a(1)) for n>1.

D-finite with recurrence: +(n+2)*a(n) +(-2*n-1)*a(n-1) +(n-1)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jan 25 2020

Example

G.f. = x + x^2 + x^3 + 3*x^4 + 7*x^5 + 13*x^6 + 29*x^7 + 71*x^8 + 163*x^9 + ...

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( 2 * x / (1 - x + sqrt(1 - 2*x + x^2 - 8*x^3 + x * O(x^n))), n))};

Crossrefs

Cf. A025249.

Sequence in context: A283323 A260022 A134270 * A025249 A147098 A109291

Adjacent sequences: A091562 A091563 A091564 * A091566 A091567 A091568

Keyword

nonn

Author

Michael Somos, Jan 21 2004

Status

approved