%I #24 Apr 12 2023 11:03:44

%S 1,3,9,30,99,330,1098,3660,12195,40650,135486,451620,1505358,5017860,

%T 16726068,55753560,185844771,619482570,2064940470,6883134900,

%U 22943778138,76479260460,254930851404,849769504680,2832564956814

%N A Catalan-related transform of 3^n.

%C 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 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 Vincenzo Librandi, <a href="/A102898/b102898.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 G.f.: 2*x/(3*sqrt(1-4*x^2) + 2*x - 3).

%F 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 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 a(n) ~ 2^(n+2) * 5^(n-1) / 3^n. - _Vaclav Kotesovec_, Feb 01 2014

%t CoefficientList[Series[2*x/(3*Sqrt[1-4*x^2]+2*x-3), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 01 2014 *)

%o (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 (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 Cf. A000108, A098615, A100087.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jan 17 2005