A136576 Series reversion of x*c(x)/(1 - 2*x), c(x) the g.f. of A000108.

0, 1, -3, 10, -36, 136, -532, 2136, -8752, 36448, -153824, 656448, -2827904, 12281088, -53709632, 236337536, -1045603072, 4648306176, -20753783296, 93022530560, -418415228928, 1888065744896, -8544699844608, 38774062837760

Offset: 0

Author

Paul Barry, Jan 08 2008

Keywords

• easy
• sign

Comments

Hankel transform of a(n+1) is A136577 (conjecture).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A069731, A071356.

Programs

Mathematica

CoefficientList[Series[(Sqrt[1+4*x-4*x^2]+4*x^2-2*x-1)/(8*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 22 2014 *)

PARI

x='x+O('x^50); concat([0], Vec((sqrt(1+4*x-4*x^2)+4*x^2-2*x-1)/(8*x^2))) \\ G. C. Greubel, Mar 21 2017

Formula

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

D-finite with recurrence (n+2)*a(n) + 2*(2*n+1)*a(n-1) + 4*(1-n)*a(n-2) = 0. - R. J. Mathar, Dec 11 2011

a(n) ~ (-1)^(n+1) * (3+2*sqrt(2)) * sqrt(4-2*sqrt(2)) * 2^(n-2) * (1+sqrt(2))^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 22 2014

For n >= 1, a(n) = (-1)^(n+1) * (1/2) * A071356(n) = (-1)^(n+1) * Sum_{k = 0..floor(n/2)} binomial(n, 2*k)*Catalan(k)*2^(n-k-1). The recurrence given above follows from this using the WZ algorithm. - Peter Bala, Apr 28 2024