A025751 6th-order Patalan numbers (generalization of Catalan numbers).
1, 1, 15, 330, 8415, 232254, 6735366, 202060980, 6213375135, 194685754230, 6191006984514, 199237861137996, 6475230486984870, 212188322111965740, 7002214629694869420, 232473525705869664744, 7758803920433400060831, 260148131449825766745510, 8758320425477467480432170
Offset: 0
Keywords
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[(7 - (1 - 36*x)^(1/6))/6, {x, 0, 20}], x] (* Vincenzo Librandi, Dec 29 2012 *) a[n_] := 36^(n-1) * Pochhammer[5/6, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *) -
Maxima
a[0]:1$ a[1]:1$ a[n]:=(6/n)*(6*n-7)*a[n-1]$ makelist(a[n],n,0,1000); /* Tani Akinari, Aug 03 2014 */
Formula
G.f.: (7-(1-36*x)^(1/6))/6.
a(n) = 6^(n-1)*5*A034787(n-1)/n!, n >= 2, where 5*A034787(n-1)=(6*n-7)(!^6) = Product_{j=2..n} (6*j - 7). - Wolfdieter Lang.
a(n) ~ 36^(n-1) / (Gamma(5/6) * n^(7/6)). - Amiram Eldar, Aug 20 2025