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).

The formula a(n, m) = m^(2*n-2) * Pochhammer((m-1)/m, n-1)/n!, with a(0, m) = 1, leads to the generating function f(m, x) = Sum_{n>=0} a(n,m)*x^n = (1/m)*(m+1 - (1 - m^2*x)^(1/m)). The set of sequences A019590, this sequence, A025749 to A025755, and A120588 are obtained when 1 <= m <= 10. - G. C. Greubel, Oct 30 2025

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) = 2 * 3^(n-1) * 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

For n >= 1, a(n) = Integral_{x = 0..9} x^n * w(x) dx, where w(x) = 1/(2*sqrt(3)*Pi) * x^(2/3)*(9 - x)^(1/3)/x^2. - Peter Bala, Oct 14 2024

a(n) ~ 9^(n-1) / (Gamma(2/3) * n^(4/3)). - Amiram Eldar, Aug 20 2025

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 *)
a[n_] := 9^(n-1) * Pochhammer[2/3, n-1]/n!; a[0] = 1; Array[a, 25, 0] (* Amiram Eldar, Aug 20 2025 *)

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
(SageMath)
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
(SageMath)
def A025748(n): return 1 if n==0 else 3^(2*n-2)*rising_factorial(2/3, n-1)/factorial(n)
[A025748(n) for n in range(41)] # G. C. Greubel, Oct 30 2025

Crossrefs

Apart from the initial 1, identical to A097188.

Cf. A004989, A005130, A034000, A034164, A034171, A073006.

Cf. A019590, A025749, A025750, A025751, A025752, A025753, A025754, A025755, A120588.

Sequence in context: A370186 A361843 A097188 * A396415 A366085 A394159

Adjacent sequences: A025745 A025746 A025747 * A025749 A025750 A025751

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved