A097188 G.f. A(x) satisfies A057083(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097186.

1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260, 156636118477018620

Offset: 0

Author

Paul D. Hanna, Aug 03 2004

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4, eq.(23) for l=4.
• Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
• Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Crossrefs

Cf. A004988, A025748, A034164, A057083, A097186.

Essentially identical to A025748.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1 - (1-9*x)^(1/3))/(3*x) )); // G. C. Greubel, Sep 17 2019

Maple

seq(coeff(series((1-(1-9*x)^(1/3))/(3*x), x, n+2), x, n), n = 0..25); # G. C. Greubel, Sep 17 2019

Mathematica

Table[FullSimplify[9^n * Gamma[n+2/3] / ((n+1) * Gamma[2/3] * Gamma[n+1])],{n,0,20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1-(1 - 9 x)^(1/3))/(3 x), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 10 2014 *)

PARI

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

Sage

def A097188_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1 - (1-9*x)^(1/3))/(3*x)).list()
A097188_list(25) # G. C. Greubel, Sep 17 2019

Formula

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

G.f.: A(x) = (1/x)*(series reversion of x/A057083(x)).

a(n) = A004988(n)/(n+1).

a(n) = A025748(n+1).

a(n) = 3*A034164(n-1) for n>=1.

x*A(x) is the compositional inverse of x-3*x^2+3*x^3. - Ira M. Gessel, Feb 18 2012

a(n) = 1/(n+1) * Sum_{k=1..n} binomial(k,n-k) * 3^(k)*(-1)^(n-k) * binomial(n+k,n), if n>0; a(0)=1. - Vladimir Kruchinin, Feb 07 2011

Conjecture: (n+1)*a(n) +3*(-3*n+1)*a(n-1)=0. - R. J. Mathar, Nov 16 2012

a(n) = 9^n * Gamma(n+2/3) / ((n+1) * Gamma(2/3) * Gamma(n+1)). - Vaclav Kotesovec, Feb 09 2014

Sum_{n>=0} 1/a(n) = 21/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - Amiram Eldar, Dec 02 2022