|
%I #52 Dec 02 2022 07:05:05
%S 1,3,15,90,594,4158,30294,227205,1741905,13586859,107459703,859677624,
%T 6943550040,56540336040,463630755528,3824953733106,31724616256938,
%U 264371802141150,2212374554760150,18583946259985260,156636118477018620
%N G.f. A(x) satisfies A057083(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097186.
%H Vincenzo Librandi, <a href="/A097188/b097188.txt">Table of n, a(n) for n = 0..200</a>
%H 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, eq.(23) for l=4.
%H 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 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 G.f.: A(x) = (1 - (1-9*x)^(1/3))/(3*x).
%F G.f.: A(x) = (1/x)*(series reversion of x/A057083(x)).
%F a(n) = A004988(n)/(n+1).
%F a(n) = A025748(n+1).
%F a(n) = 3*A034164(n-1) for n>=1.
%F x*A(x) is the compositional inverse of x-3*x^2+3*x^3. - _Ira M. Gessel_, Feb 18 2012
%F a(n) = 1/(n+1) * Sum_{k=1..n} binomial(k,n-k) * 3^(k)*(-1)^(n-k) * binomial(n+k,n), if n>0; a(0)=1. - _Vladimir Kruchinin_, Feb 07 2011
%F Conjecture: (n+1)*a(n) +3*(-3*n+1)*a(n-1)=0. - _R. J. Mathar_, Nov 16 2012
%F a(n) = 9^n * Gamma(n+2/3) / ((n+1) * Gamma(2/3) * Gamma(n+1)). - _Vaclav Kotesovec_, Feb 09 2014
%F Sum_{n>=0} 1/a(n) = 21/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - _Amiram Eldar_, Dec 02 2022
%p seq(coeff(series((1-(1-9*x)^(1/3))/(3*x), x, n+2), x, n), n = 0..25); # _G. C. Greubel_, Sep 17 2019
%t Table[FullSimplify[9^n * Gamma[n+2/3] / ((n+1) * Gamma[2/3] * Gamma[n+1])],{n,0,20}] (* _Vaclav Kotesovec_, Feb 09 2014 *)
%t CoefficientList[Series[(1-(1 - 9 x)^(1/3))/(3 x), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 10 2014 *)
%o (PARI) a(n)=polcoeff((1-(1-9*x+x^2*O(x^n))^(1/3))/(3*x),n,x)
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1 - (1-9*x)^(1/3))/(3*x) )); // _G. C. Greubel_, Sep 17 2019
%o (Sage)
%o def A097188_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P((1 - (1-9*x)^(1/3))/(3*x)).list()
%o A097188_list(25) # _G. C. Greubel_, Sep 17 2019
%Y Cf. A004988, A025748, A034164, A057083, A097186.
%Y Essentially identical to A025748.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 03 2004
| 