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A097194 Row sums of triangle A097190, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097191(y)^(n+1), where R_n(1/3) = 9^n for all n>=0.

%I A097194 #8 Sep 08 2022 08:45:14
%S A097194 1,25,649,17065,451621,11998801,319623445,8530126057,227974775239,
%T A097194 6099550226965,163340461497907,4377292845062689,117376545230379631,
%U A097194 3149059523347103293,84522568856319875179,2269506752111508954553
%N A097194 Row sums of triangle A097190, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097191(y)^(n+1), where R_n(1/3) = 9^n for all n>=0.
%H A097194 G. C. Greubel, <a href="/A097194/b097194.txt">Table of n, a(n) for n = 0..690</a>
%F A097194 G.f.: A(x) = 3/((1-27*x) + 2*(1-27*x)^(8/9)).
%F A097194 G.f.: A(x, y) = A097192(x)/(1 - x*A097193(x)).
%p A097194 seq(coeff(series(3/((1-27*x) +2*(1-27*x)^(8/9)), x, n+1), x, n), n = 0 ..20); # _G. C. Greubel_, Sep 17 2019
%t A097194 CoefficientList[Series[3/((1-27*x) +2*(1-27*x)^(8/9)), {x,0,20}], x] (* _G. C. Greubel_, Sep 17 2019 *)
%o A097194 (PARI) a(n)=polcoeff(3/((1-27*x) + 2*(1-27*x+x*O(x^n))^(8/9)),n,x)
%o A097194 (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 3/((1-27*x) +2*(1-27*x)^(8/9)) )); // _G. C. Greubel_, Sep 17 2019
%o A097194 (Sage)
%o A097194 def A097194_list(prec):
%o A097194     P.<x> = PowerSeriesRing(QQ, prec)
%o A097194     return P(3/((1-27*x) +2*(1-27*x)^(8/9))).list()
%o A097194 A097194_list(20) # _G. C. Greubel_, Sep 17 2019
%Y A097194 Cf. A097190.
%K A097194 nonn
%O A097194 0,2
%A A097194 _Paul D. Hanna_, Aug 03 2004