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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Transform of 1/(1-3x) under the mapping g(x)->g(xc(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, M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)

FORMULA

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

a(0)=1, a(n)=sum{k=0..n, k*binomial(n-1, (n-k)/2)(1+(-1)^(n-k))3^k/(n+k)}, n>0.

Conjecture: 3*n*a(n) -10*n*a(n-1) +12*(3-n)*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 *)

CROSSREFS

Cf. A100087, A098615.

Sequence in context: A199137 A089978 A052906 * A050181 A275690 A089931

Adjacent sequences:  A102895 A102896 A102897 * A102899 A102900 A102901

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jan 17 2005

STATUS

approved

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Last modified December 12 20:12 EST 2019. Contains 329961 sequences. (Running on oeis4.)