A126986 Expansion of 1/(1+4*x*c(x)), c(x) the g.f. of Catalan numbers A000108.

1, -4, 12, -40, 124, -408, 1272, -4176, 13020, -42808, 133096, -439344, 1358872, -4514800, 13853040, -46469280, 140945820, -479312760, 1430085000, -4958382960, 14453014920, -51500944080, 145230007440, -537922074720, 1446902948184, -5662012752048, 14228883685392

Offset: 0

Author

Philippe Deléham, Mar 21 2007

Keywords

• sign

Comments

Hankel transform is (-4)^n.

For n>=37, all terms are negative. - Vaclav Kotesovec, May 30 2019

Links

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

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(3 - 2*Sqrt(1-4*x)) )); // G. C. Greubel, May 29 2019

Maple

c:=(1-sqrt(1-4*x))/2/x: ser:=series(1/(1+4*x*c),x=0,30): seq(coeff(ser,x,n),n=0..27); # Emeric Deutsch, Mar 23 2007

Mathematica

CoefficientList[Series[1/(3-2*Sqrt[1-4*x]), {x,0,30}], x] (* G. C. Greubel, May 29 2019 *)

PARI

my(x='x+O('x^30)); Vec(1/(3-2*sqrt(1-4*x))) \\ G. C. Greubel, May 29 2019

Sage

(1/(3-2*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)*(-5)^k.

G.f.: 1/(3 - 2*sqrt(1-4*x)). - G. C. Greubel, May 29 2019

a(n) ~ -4^n / (9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, May 30 2019

D-finite with recurrence -5*n*a(n) +2*(2*n-15)*a(n-1) +32*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024

Extensions

More terms from Emeric Deutsch, Mar 23 2007