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A102898 A Catalan-related transform of 3^n.

%I A102898 #24 Apr 12 2023 11:03:44
%S A102898 1,3,9,30,99,330,1098,3660,12195,40650,135486,451620,1505358,5017860,
%T A102898 16726068,55753560,185844771,619482570,2064940470,6883134900,
%U A102898 22943778138,76479260460,254930851404,849769504680,2832564956814
%N A102898 A Catalan-related transform of 3^n.
%C A102898 Transform of 1/(1-3*x) under the mapping g(x) -> g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108. The inverse transform is h(x) -> h(x/(1+x^2)).
%D A102898 Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.
%H A102898 Vincenzo Librandi, <a href="/A102898/b102898.txt">Table of n, a(n) for n = 0..1000</a>
%H A102898 S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017).
%F A102898 G.f.: 2*x/(3*sqrt(1-4*x^2) + 2*x - 3).
%F A102898 a(n) = Sum_{k=0..n} k*binomial(n-1, (n-k)/2)*(1 + (-1)^(n-k))*3^k/(n+k), n > 0, with a(0) = 1.
%F A102898 3*n*a(n) - 10*n*a(n-1) - 12*(n-3)*a(n-2) + 40*(n-3)*a(n-3) = 0. - _R. J. Mathar_, Sep 21 2012
%F A102898 a(n) ~ 2^(n+2) * 5^(n-1) / 3^n. - _Vaclav Kotesovec_, Feb 01 2014
%t A102898 CoefficientList[Series[2*x/(3*Sqrt[1-4*x^2]+2*x-3), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 01 2014 *)
%o A102898 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2*x/(3*Sqrt(1-4*x^2)+2*x-3) )); // _G. C. Greubel_, Jul 08 2022
%o A102898 (SageMath) [1]+[2*sum(k*binomial(n-1, (n-k)//2)*((n-k+1)%2)*3^k/(n+k) for k in (0..n)) for n in (1..40)] # _G. C. Greubel_, Jul 08 2022
%Y A102898 Cf. A000108, A098615, A100087.
%K A102898 easy,nonn
%O A102898 0,2
%A A102898 _Paul Barry_, Jan 17 2005