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A025748 3rd-order Patalan numbers (generalization of Catalan numbers).

%I A025748 #76 Aug 20 2025 08:58:13
%S A025748 1,1,3,15,90,594,4158,30294,227205,1741905,13586859,107459703,
%T A025748 859677624,6943550040,56540336040,463630755528,3824953733106,
%U A025748 31724616256938,264371802141150,2212374554760150,18583946259985260,156636118477018620,1324287183487521060
%N A025748 3rd-order Patalan numbers (generalization of Catalan numbers).
%C A025748 G.f. (with a(0)=0) is series reversion of x - 3*x^2 + 3*x^3.
%C A025748 The Hankel transform of a(n) is A005130(n) * 3^binomial(n,2).
%H A025748 Vincenzo Librandi, <a href="/A025748/b025748.txt">Table of n, a(n) for n = 0..1000</a>
%H A025748 I. M. Gessel and G. Xin, <a href="http://arxiv.org/abs/math/0505217">The generating function of ternary trees and continued fractions</a>, arXiv:math/0505217 [math.CO], 2005, eq. (5.1).
%H A025748 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
%H A025748 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 A025748 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 A025748 From _Wolfdieter Lang_: (Start)
%F A025748 G.f.: (4 - (1-9*x)^(1/3))/3.
%F A025748 a(n) = 3^(n-1) * 2 * A034000(n-1)/n!, n >= 2.
%F A025748 a(n) = 3 * A034164(n-2), n >= 2. (End)
%F A025748 D-finite with recurrence n*a(n) + 3*(4-3*n)*a(n-1) = 0, n >= 2. - _R. J. Mathar_, Oct 29 2012
%F A025748 For n>0, a(n) = 9^(n-1) * Gamma(n-1/3) / (n * Gamma(2/3) * Gamma(n)). - _Vaclav Kotesovec_, Feb 09 2014
%F A025748 For n > 0, a(n) = 3^(2*n-1)*(-1)^(n+1)*binomial(1/3, n). - _Peter Bala_, Mar 01 2022
%F A025748 Sum_{n>=0} 1/a(n) = 37/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - _Amiram Eldar_, Dec 02 2022
%F A025748 For n >= 1, a(n) = Integral_{x = 0..9} x^n * w(x) dx, where w(x) = 1/(2*sqrt(3)*Pi) *  x^(2/3)*(9 - x)^(1/3)/x^2. - _Peter Bala_, Oct 14 2024
%F A025748 a(n) ~ 9^(n-1) / (Gamma(2/3) * n^(4/3)). - _Amiram Eldar_, Aug 20 2025
%p A025748 A025748 :=proc(n)
%p A025748         local x;
%p A025748         coeftayl(4-(1-9*x)^(1/3),x=0,n) ;
%p A025748         %/3 ;
%p A025748 end proc: # _R. J. Mathar_, Nov 01 2012
%t A025748 CoefficientList[Series[(4-Power[1-9x, (3)^-1])/3,{x,0,25}],x] (* _Harvey P. Dale_, Nov 14 2011 *)
%t A025748 Flatten[{1,Table[FullSimplify[9^(n-1) * Gamma[n-1/3] / (n * Gamma[2/3] * Gamma[n])],{n,1,25}]}] (* _Vaclav Kotesovec_, Feb 09 2014 *)
%t A025748 a[n_] := 9^(n-1) * Pochhammer[2/3, n-1]/n!; a[0] = 1; Array[a, 25, 0] (* _Amiram Eldar_, Aug 20 2025 *)
%o A025748 (PARI) a(n)=if(n<1,n==0,polcoeff(serreverse(x-3*x^2+3*x^3+x*O(x^n)),n))
%o A025748 (Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (4 - (1-9*x)^(1/3))/3 )); // _G. C. Greubel_, Sep 17 2019
%o A025748 (Sage)
%o A025748 def A025748_list(prec):
%o A025748     P.<x> = PowerSeriesRing(QQ, prec)
%o A025748     return P((4 - (1-9*x)^(1/3))/3).list()
%o A025748 A025748_list(25) # _G. C. Greubel_, Sep 17 2019
%Y A025748 Apart from the initial 1, identical to A097188.
%Y A025748 Cf. A004989, A005130, A034000, A034164, A073006,
%K A025748 nonn,easy
%O A025748 0,3
%A A025748 _Olivier Gérard_