A126985 Expansion of 1/(1+8*x*c(x)), c(x) the g.f. of Catalan numbers A000108.

1, -8, 56, -400, 2840, -20208, 143664, -1021728, 7265240, -51665200, 367392656, -2612584928, 18578329456, -132112749920, 939467783520, -6680662171200, 47506922377560, -337827035002800, 2402325467002320, -17083203745473120, 121480558396908240, -863861754435010080

Offset: 0

Author

Philippe Deléham, Mar 21 2007

Keywords

• sign

Comments

Hankel transform is (-8)^n.

Catalan transform of (-1)^n*A001018(n). - R. J. Mathar, Nov 11 2008

Links

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

Crossrefs

Cf. A000108, A001018, A039599.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(10-8*Sqrt(1-4*x)) )); // G. C. Greubel, May 28 2019

Maple

c:=(1-sqrt(1-4*x))/2/x: ser:=series(1/(1+8*x*c),x=0,25): seq(coeff(ser,x,n),n=0..21); # Emeric Deutsch, Mar 24 2007

Mathematica

CoefficientList[Series[2/(10-8*Sqrt[1-4*x]), {x,0,30}], x] (* G. C. Greubel, May 28 2019 *)

PARI

my(x='x+O('x^30)); Vec(2/(10-8*sqrt(1-4*x))) \\ G. C. Greubel, May 28 2019

Sage

(2/(10-8*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 28 2019

Formula

a(n) = Sum_{k=0..n} A039599(n,k)*(-9)^k.

G.f.: 2/(10 - 8*sqrt(1-4*x)). - G. C. Greubel, May 28 2019

D-finite with recurrence 9*n*a(n) +2*(14*n+27)*a(n-1) +128*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024

Extensions

More terms from Emeric Deutsch, Mar 24 2007