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

Author

Olivier Gérard

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Wolfdieter Lang, On generalizations of 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.

Crossrefs

Cf. A035022.

Programs

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 *)

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