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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-3x^2+3x^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.CO/0505217, eq. (5.1).

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

T. M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880, 2014 and J. Int. Seq. 18 (2015) # 15.3.3.

FORMULA

G.f.: (4-(1-9x)^(1/3))/3. a(n)=3^(n-1)*2*A034000(n-1)/n!, n >= 2 and a(n)=3*A034164(n-2), n >= 2. - Wolfdieter Lang

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, 20}], 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, 20}]}] (* 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))

CROSSREFS

Apart from the initial 1, identical to A097188.

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 September 19 09:18 EDT 2018. Contains 315192 sequences. (Running on oeis4.)