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A102898
A Catalan-related transform of 3^n.
7
1, 3, 9, 30, 99, 330, 1098, 3660, 12195, 40650, 135486, 451620, 1505358, 5017860, 16726068, 55753560, 185844771, 619482570, 2064940470, 6883134900, 22943778138, 76479260460, 254930851404, 849769504680, 2832564956814
OFFSET
0,2
COMMENTS
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)).
REFERENCES
Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.
FORMULA
G.f.: 2*x/(3*sqrt(1-4*x^2) + 2*x - 3).
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.
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
a(n) ~ 2^(n+2) * 5^(n-1) / 3^n. - Vaclav Kotesovec, Feb 01 2014
MATHEMATICA
CoefficientList[Series[2*x/(3*Sqrt[1-4*x^2]+2*x-3), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)
PROG
(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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jan 17 2005
STATUS
approved