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A025748 3rd order Patalan numbers (generalization of Catalan numbers). 14
1, 1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

G.f. (with a(0)=0) is series reversion of x - 3*x^2 + 3*x^3.

The Hankel transform of a(n) is A005130(n) * 3^binomial(n,2).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

I. M. Gessel, G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005, eq. (5.1).

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

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 [math.CO], 2014 and J. Int. Seq. 18 (2015) # 15.3.3.

FORMULA

From Wolfdieter Lang: (Start)

G.f.: (4 - (1-9*x)^(1/3))/3.

a(n) = 3^(n-1) * 2 * A034000(n-1)/n!, n >= 2.

a(n) = 3 * A034164(n-2), n >= 2. (End)

n*a(n) +3*(4-3*n)*a(n-1) = 0, n>=2. - R. J. Mathar, Oct 29 2012

For n>0, a(n) = 9^(n-1) * Gamma(n-1/3) / (n * Gamma(2/3) * Gamma(n)). - Vaclav Kotesovec, Feb 09 2014

MAPLE

A025748 :=proc(n)

        local x;

        coeftayl(4-(1-9*x)^(1/3), x=0, n) ;

        %/3 ;

end proc: # R. J. Mathar, Nov 01 2012

MATHEMATICA

CoefficientList[Series[(4-Power[1-9x, (3)^-1])/3, {x, 0, 25}], x] (* Harvey P. Dale, Nov 14 2011 *)

Flatten[{1, Table[FullSimplify[9^(n-1) * Gamma[n-1/3] / (n * Gamma[2/3] * Gamma[n])], {n, 1, 25}]}] (* Vaclav Kotesovec, Feb 09 2014 *)

PROG

(PARI) a(n)=if(n<1, n==0, polcoeff(serreverse(x-3*x^2+3*x^3+x*O(x^n)), n))

(MAGMA) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (4 - (1-9*x)^(1/3))/3 )); // G. C. Greubel, Sep 17 2019

(Sage)

def A025748_list(prec):

    P.<x> = PowerSeriesRing(QQ, prec)

    return P((4 - (1-9*x)^(1/3))/3).list()

A025748_list(25) # G. C. Greubel, Sep 17 2019

CROSSREFS

Apart from the initial 1, identical to A097188.

Cf. A005130, A034000, A034164.

Sequence in context: A205576 A173695 A255688 * A097188 A271930 A201953

Adjacent sequences:  A025745 A025746 A025747 * A025749 A025750 A025751

KEYWORD

nonn

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified December 2 23:36 EST 2020. Contains 338898 sequences. (Running on oeis4.)