cp's OEIS Frontend

A025755 10th-order Patalan numbers (generalization of Catalan numbers).

%I A025755 #38 Aug 20 2025 08:58:39
%S A025755 1,1,45,2850,206625,16116750,1316201250,110936962500,9568313015625,
%T A025755 839885253593750,74749787569843750,6727480881285937500,
%U A025755 611079513383472656250,55937278532794804687500,5154220664807521289062500,477624448272163639453125000,44478776745345238924072265625
%N A025755 10th-order Patalan numbers (generalization of Catalan numbers).
%H A025755 Vincenzo Librandi, <a href="/A025755/b025755.txt">Table of n, a(n) for n = 0..200</a>
%H A025755 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 A025755 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 A025755 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 A025755 G.f.: (11-(1-100*x)^(1/10))/10.
%F A025755 a(n) = 10^(n-1)*9*A035278(n-1)/n!, n >= 2, where 9*A035278(n-1) = (10*n-11)(!^10) = Product_{j=2..n} (10*j - 11). - _Wolfdieter Lang_
%F A025755 Conjecture: n*a(n) + 10*(-10*n+11)*a(n-1) = 0. - _R. J. Mathar_, Jul 28 2014
%F A025755 a(n) = 100^(n-1)*Pochhammer(9/10, n-1)/n! for n >= 1.  Maple confirms this satisfies Mathar's conjecture for n >= 2 (it's not true for n=1). - _Robert Israel_, Oct 05 2014
%F A025755 a(n) ~ 100^(n-1) / (Gamma(9/10) * n^(11/10)). - _Amiram Eldar_, Aug 20 2025
%t A025755 CoefficientList[Series[(11 -(1 - 100*x)^(1/10))/10, {x, 0, 20}], x] (* _Vincenzo Librandi_, Dec 29 2012 *)
%t A025755 a[n_] := 100^(n-1) * Pochhammer[9/10, n-1] / n!; a[0] = 1; Array[a, 26, 0] (* _Amiram Eldar_, Aug 20 2025 *)
%Y A025755 Cf. A035278, A340725.
%K A025755 nonn
%O A025755 0,3
%A A025755 _Olivier Gérard_