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

1, 1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260, 156636118477018620, 1324287183487521060

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

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

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.

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)

D-finite with recurrence 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

For n > 0, a(n) = 3^(2*n-1)*(-1)^(n+1)*binomial(1/3, n). - Peter Bala, Mar 01 2022

Sum_{n>=0} 1/a(n) = 37/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - Amiram Eldar, Dec 02 2022

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: A370186 A361843 A097188 * A366085 A271930 A201953

Adjacent sequences: A025745 A025746 A025747 * A025749 A025750 A025751

Keyword

nonn

Author

Olivier Gérard

Status

approved