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Main diagonal 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.
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%I #9 Sep 08 2022 08:45:14

%S 1,24,612,15912,417690,11027016,292215924,7764594552,206732329947,

%T 5512862131920,147193418922264,3934078651195056,105236603919467748,

%U 2817102935690367408,75458114348849127000,2022277464549156603600

%N Main diagonal 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 G. C. Greubel, <a href="/A097192/b097192.txt">Table of n, a(n) for n = 0..695</a>

%F G.f.: A(x) = 1/(1-27*x)^(8/9).

%F a(n) = (n+1)*A097193(n).

%F Conjecture: n*a(n) +3*(1-9*n)*a(n-1) = 0. - _R. J. Mathar_, Nov 16 2012

%p seq(coeff(series(1/(1-27*x)^(8/9), x, n+1), x, n), n = 0 ..20); # _G. C. Greubel_, Sep 17 2019

%t CoefficientList[Series[(1-27*x)^(-8/9), {x,0,20}], x] (* _G. C. Greubel_, Sep 17 2019 *)

%o (PARI) a(n)=polcoeff(1/(1-27*x+x*O(x^n))^(8/9),n,x)

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 1/(1-27*x)^(8/9) )); // _G. C. Greubel_, Sep 17 2019

%o (Sage)

%o def A097192_list(prec):

%o P.<x> = PowerSeriesRing(QQ, prec)

%o return P(1/(1-27*x)^(8/9)).list()

%o A097192_list(20) # _G. C. Greubel_, Sep 17 2019

%Y Cf. A097190, A097191, A097193.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 03 2004