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A097188
G.f. A(x) satisfies A057083(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097186.
14
1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260, 156636118477018620
OFFSET
0,2
LINKS
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
Essentially identical to A025748.
Sequence in context: A255688 A370186 A361843 * A025748 A366085 A271930
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 03 2004
STATUS
approved