A126987 Expansion of 1/(1+5*x*c(x)), c(x) the g.f. of Catalan numbers A000108.

1, -5, 20, -85, 350, -1470, 6090, -25485, 105830, -442150, 1838240, -7673330, 31923220, -133186760, 554325750, -2311919325, 9624918150, -40133290350, 167114005800, -696706389750, 2901470571300, -12094930814100, 50375156502900, -209972720898450, 874600454065500

Offset: 0

Author

Philippe Deléham, Mar 21 2007

Keywords

• sign

Comments

Hankel transform is (-5)^n.

Links

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

Crossrefs

Cf. A000108, A039599.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(7 - 5*Sqrt(1-4*x)) )); // G. C. Greubel, May 29 2019

Maple

c:=(1-sqrt(1-4*x))/2/x: ser:=series(1/(1+5*x*c),x=0,27): seq(coeff(ser,x,n),n=0..24); # Emeric Deutsch, Mar 23 2007

Mathematica

CoefficientList[Series[2/(7-5*Sqrt[1-4*x]), {x,0,30}], x] (* G. C. Greubel, May 29 2019 *)

PARI

my(x='x+O('x^30)); Vec(2/(7-5*sqrt(1-4*x))) \\ G. C. Greubel, May 29 2019

Sage

(2/(7-5*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 29 2019

Formula

a(n) = Sum_{k=0..n} A039599(n,k)*(-6)^k.

G.f.: 2/(7 - 5*sqrt(1-4*x)). - G. C. Greubel, May 29 2019

D-finite with recurrence 6*n*a(n) +(n+36)*a(n-1) +50*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024

Extensions

More terms from Emeric Deutsch, Mar 23 2007