A097193 G.f. A(x) satisfies A097191(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097190.

1, 12, 204, 3978, 83538, 1837836, 41745132, 970574319, 22970258883, 551286213192, 13381219902024, 327839887599588, 8095123378420596, 201221638263597672, 5030540956589941800, 126392341534322287725

Offset: 0

Author

Paul D. Hanna, Aug 03 2004

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A097190, A097191, A097192.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (1-(1-27*x)^(1/9))/(3*x) )); // G. C. Greubel, Sep 17 2019

Maple

seq(coeff(series((1-(1-27*x)^(1/9))/(3*x), x, n+2), x, n), n = 0 ..20); # G. C. Greubel, Sep 17 2019

Mathematica

CoefficientList[Series[(1-(1-27*x)^(1/9))/(3*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

PARI

a(n)=polcoeff((1-(1-27*x+x^2*O(x^n))^(1/9))/(3*x),n,x)

Sage

def A097193_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1-(1-27*x)^(1/9))/(3*x)).list()
A097193_list(20) # G. C. Greubel, Sep 17 2019

Formula

G.f.: A(x) = (1-(1-27*x)^(1/9))/(3*x).

G.f.: A(x) = (1/x)*(series reversion of x/A097191(x)).

a(n) = A097192(n)/(n+1).

a(n) ~ 27^n / (Gamma(8/9) * n^(10/9)). - Vaclav Kotesovec, Feb 12 2014