A102898 A Catalan-related transform of 3^n.

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

Author

Paul Barry, Jan 17 2005

Keywords

• easy
• nonn

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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).

Crossrefs

Cf. A000108, A098615, A100087.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2*x/(3*Sqrt(1-4*x^2)+2*x-3) )); // G. C. Greubel, Jul 08 2022

Mathematica

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

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

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