A025754 9th-order Patalan numbers (generalization of Catalan numbers).

1, 1, 36, 1836, 107406, 6766578, 446594148, 30432201228, 2122646035653, 150707868531363, 10850966534258136, 790147653994615176, 58075852568604215436, 4302080463351219958836, 320812285981333831216056, 24060921448600037341204200, 1813591954188227814593266575

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Wolfdieter Lang, On generalizations of the Stirling number triangles, J. Integer Seq., 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

G.f.: (10-(1-81*x)^(1/9))/9.

a(n) = 9^(n-1)*8*A035022(n-1)/n!, n >= 2, where 8*A035022(n-1)= (9*n-10)(!^9)= Product_{j=2..n} (9*j - 10). - Wolfdieter Lang

Conjecture: n*a(n) + 9*(-9*n+10)*a(n-1) = 0. - R. J. Mathar, Jul 28 2014

a(n) ~ 81^(n-1) / (Gamma(8/9) * n^(10/9)). - Amiram Eldar, Aug 20 2025

Mathematica

CoefficientList[Series[(10-(1-81x)^(1/9))/9, {x, 0, 20}], x] (* Harvey P. Dale, Nov 29 2012 *)
a[n_] := 81^(n-1) * Pochhammer[8/9, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!( (10 - (1-9^2*x)^(1/9))/9 )); // G. C. Greubel, Oct 30 2025
(SageMath)
def A025754(n): return 1 if n==0 else 9^(2*n-2)*rising_factorial(8/9, n-1)/factorial(n)
[A025754(n) for n in range(21)] # G. C. Greubel, Oct 30 2025

Crossrefs

Cf. A025754, A035022.

Sequence in context: A391281 A391282 A151640 * A071128 A065782 A374393

Adjacent sequences: A025751 A025752 A025753 * A025755 A025756 A025757

Keyword

nonn

Author

Olivier Gérard

Status

approved