A025753 8th-order Patalan numbers (generalization of Catalan numbers).
1, 1, 28, 1120, 51520, 2555392, 132880384, 7137574912, 392566620160, 21983730728960, 1248675905404928, 71742106565083136, 4161042180774821888, 243260927491451125760, 14317643160925409116160, 847604475126784219676672, 50432466270043661070761984, 3014081513550844685170245632
Offset: 0
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.
Programs
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Mathematica
CoefficientList[Series[(9 - (1 - 64*x)^(1/8))/8, {x, 0, 20}], x] (* Vincenzo Librandi, Dec 29 2012 *) a[n_] := 64^(n-1) * Pochhammer[7/8, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)
Formula
G.f.: (9-(1-64*x)^(1/9))/8.
a(n) = 8^(n-1)*7*A034975(n-1)/n!, n >= 2, where 7*A034975(n-1)= (8*n-9)!^8 = Product_{j=2..n} (8*j - 9). - Wolfdieter Lang
a(n) ~ 64^(n-1) / (Gamma(7/8) * n^(9/8)). - Amiram Eldar, Aug 20 2025