A025752 7th-order Patalan numbers (generalization of Catalan numbers).
1, 1, 21, 637, 22295, 842751, 33429123, 1370594043, 57564949806, 2462500630590, 106872527367606, 4692675519868518, 208041948047504298, 9297874755046153626, 418404363977076913170, 18939770876029014936162, 861759574859320179595371, 39387481745040692914447251
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[(8 - (1 - 49*x)^(1/7))/7, {x, 0, 20}], x] (* Vincenzo Librandi, Dec 29 2012 *) a[n_] := 49^(n-1) * Pochhammer[6/7, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)
Formula
G.f.: (8-(1-49*x)^(1/7))/7.
a(n) = 7^(n-1)*6*A034833(n-1)/n!, n >= 2, where 6*A034833(n-1)= (7*n-8)(!^7) = Product_{j=2..n} (7*j - 8). - Wolfdieter Lang
a(n) ~ 49^(n-1) / (Gamma(6/7) * n^(8/7)). - Amiram Eldar, Aug 20 2025