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A097193 G.f. A(x) satisfies A097191(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097190.

%I A097193 #15 Sep 08 2022 08:45:14
%S A097193 1,12,204,3978,83538,1837836,41745132,970574319,22970258883,
%T A097193 551286213192,13381219902024,327839887599588,8095123378420596,
%U A097193 201221638263597672,5030540956589941800,126392341534322287725
%N A097193 G.f. A(x) satisfies A097191(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097190.
%H A097193 Vincenzo Librandi, <a href="/A097193/b097193.txt">Table of n, a(n) for n = 0..200</a>
%F A097193 G.f.: A(x) = (1-(1-27*x)^(1/9))/(3*x).
%F A097193 G.f.: A(x) = (1/x)*(series reversion of x/A097191(x)).
%F A097193 a(n) = A097192(n)/(n+1).
%F A097193 a(n) ~ 27^n / (Gamma(8/9) * n^(10/9)). - _Vaclav Kotesovec_, Feb 12 2014
%p A097193 seq(coeff(series((1-(1-27*x)^(1/9))/(3*x), x, n+2), x, n), n = 0 ..20); # _G. C. Greubel_, Sep 17 2019
%t A097193 CoefficientList[Series[(1-(1-27*x)^(1/9))/(3*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)
%o A097193 (PARI) a(n)=polcoeff((1-(1-27*x+x^2*O(x^n))^(1/9))/(3*x),n,x)
%o A097193 (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (1-(1-27*x)^(1/9))/(3*x) )); // _G. C. Greubel_, Sep 17 2019
%o A097193 (Sage)
%o A097193 def A097193_list(prec):
%o A097193     P.<x> = PowerSeriesRing(QQ, prec)
%o A097193     return P((1-(1-27*x)^(1/9))/(3*x)).list()
%o A097193 A097193_list(20) # _G. C. Greubel_, Sep 17 2019
%Y A097193 Cf. A097190, A097191, A097192.
%K A097193 nonn
%O A097193 0,2
%A A097193 _Paul D. Hanna_, Aug 03 2004