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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. 9
1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260, 156636118477018620 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. eq.(23) for l=4.

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

T. M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880, 2014

T. M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015) # 15.3.3

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

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 *)

PROG

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

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

(Sage)

def A097188_list(prec):

    P.<x> = PowerSeriesRing(QQ, prec)

    return P((1 - (1-9*x)^(1/3))/(3*x)).list()

A097188_list(25) # G. C. Greubel, Sep 17 2019

CROSSREFS

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

Essentially identical to A025748.

Sequence in context: A173695 A255688 A025748 * A271930 A201953 A185369

Adjacent sequences:  A097185 A097186 A097187 * A097189 A097190 A097191

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 03 2004

STATUS

approved

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Last modified November 21 14:18 EST 2019. Contains 329371 sequences. (Running on oeis4.)