OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4, eq.(23) for l=4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.
FORMULA
G.f.: A(x) = (1 - (1-9*x)^(1/3))/(3*x).
G.f.: A(x) = (1/x)*(series reversion of x/A057083(x)).
a(n) = A004988(n)/(n+1).
a(n) = A025748(n+1).
a(n) = 3*A034164(n-1) for n>=1.
x*A(x) is the compositional inverse of x-3*x^2+3*x^3. - Ira M. Gessel, Feb 18 2012
a(n) = 1/(n+1) * Sum_{k=1..n} binomial(k,n-k) * 3^(k)*(-1)^(n-k) * binomial(n+k,n), if n>0; a(0)=1. - Vladimir Kruchinin, Feb 07 2011
Conjecture: (n+1)*a(n) +3*(-3*n+1)*a(n-1)=0. - R. J. Mathar, Nov 16 2012
a(n) = 9^n * Gamma(n+2/3) / ((n+1) * Gamma(2/3) * Gamma(n+1)). - Vaclav Kotesovec, Feb 09 2014
Sum_{n>=0} 1/a(n) = 21/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - Amiram Eldar, Dec 02 2022
MAPLE
seq(coeff(series((1-(1-9*x)^(1/3))/(3*x), x, n+2), x, n), n = 0..25); # G. C. Greubel, Sep 17 2019
MATHEMATICA
Table[FullSimplify[9^n * Gamma[n+2/3] / ((n+1) * Gamma[2/3] * Gamma[n+1])], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1-(1 - 9 x)^(1/3))/(3 x), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 10 2014 *)
PROG
(PARI) a(n)=polcoeff((1-(1-9*x+x^2*O(x^n))^(1/3))/(3*x), n, x)
(Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1 - (1-9*x)^(1/3))/(3*x) )); // G. C. Greubel, Sep 17 2019
(Sage)
def A097188_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P((1 - (1-9*x)^(1/3))/(3*x)).list()
A097188_list(25) # G. C. Greubel, Sep 17 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 03 2004
STATUS
approved