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A025751 6th-order Patalan numbers (generalization of Catalan numbers).

%I A025751 #36 Aug 20 2025 08:58:22
%S A025751 1,1,15,330,8415,232254,6735366,202060980,6213375135,194685754230,
%T A025751 6191006984514,199237861137996,6475230486984870,212188322111965740,
%U A025751 7002214629694869420,232473525705869664744,7758803920433400060831,260148131449825766745510,8758320425477467480432170
%N A025751 6th-order Patalan numbers (generalization of Catalan numbers).
%H A025751 Vincenzo Librandi, <a href="/A025751/b025751.txt">Table of n, a(n) for n = 0..200</a>
%H A025751 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
%H A025751 Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%H A025751 Thomas M. Richardson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Richardson/rich2.html">The Super Patalan Numbers</a>, J. Int. Seq. 18 (2015), Article 15.3.3; <a href="http://arxiv.org/abs/1410.5880">arXiv preprint</a>, arXiv:1410.5880 [math.CO], 2014.
%F A025751 G.f.: (7-(1-36*x)^(1/6))/6.
%F A025751 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_.
%F A025751 a(n) ~ 36^(n-1) / (Gamma(5/6) * n^(7/6)). - _Amiram Eldar_, Aug 20 2025
%t A025751 CoefficientList[Series[(7 - (1 - 36*x)^(1/6))/6, {x, 0, 20}], x] (* _Vincenzo Librandi_, Dec 29 2012 *)
%t A025751 a[n_] := 36^(n-1) * Pochhammer[5/6, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* _Amiram Eldar_, Aug 20 2025 *)
%o A025751 (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 */
%Y A025751 Cf. A034787, A203145.
%K A025751 nonn
%O A025751 0,3
%A A025751 _Olivier Gérard_