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A025748 3rd order Patalan numbers (generalization of Catalan numbers). 14

%I

%S 1,1,3,15,90,594,4158,30294,227205,1741905,13586859,107459703,

%T 859677624,6943550040,56540336040,463630755528,3824953733106,

%U 31724616256938,264371802141150,2212374554760150,18583946259985260

%N 3rd order Patalan numbers (generalization of Catalan numbers).

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

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

%H Vincenzo Librandi, <a href="/A025748/b025748.txt">Table of n, a(n) for n = 0..1000</a>

%H I. M. Gessel, G. Xin, <a href="http://arxiv.org/abs/math/0505217">The generating function of ternary trees and continued fractions</a>, arXiv:math/0505217 [math.CO], 2005, eq. (5.1).

%H W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%H Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.

%H T. M. Richardson, <a href="http://arxiv.org/abs/1410.5880">The Super Patalan Numbers</a>, arXiv preprint arXiv:1410.5880 [math.CO], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Richardson/rich2.html">J. Int. Seq. 18 (2015) # 15.3.3</a>.

%F From _Wolfdieter Lang_: (Start)

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

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

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

%F n*a(n) +3*(4-3*n)*a(n-1) = 0, n>=2. - _R. J. Mathar_, Oct 29 2012

%F For n>0, a(n) = 9^(n-1) * Gamma(n-1/3) / (n * Gamma(2/3) * Gamma(n)). - _Vaclav Kotesovec_, Feb 09 2014

%p A025748 :=proc(n)

%p local x;

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

%p %/3 ;

%p end proc: # _R. J. Mathar_, Nov 01 2012

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

%t 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 *)

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

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

%o (Sage)

%o def A025748_list(prec):

%o P.<x> = PowerSeriesRing(QQ, prec)

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

%o A025748_list(25) # _G. C. Greubel_, Sep 17 2019

%Y Apart from the initial 1, identical to A097188.

%Y Cf. A005130, A034000, A034164.

%K nonn

%O 0,3

%A _Olivier Gérard_

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Last modified July 15 02:31 EDT 2020. Contains 335762 sequences. (Running on oeis4.)