A097191 - OEIS

%I #12 Sep 08 2022 08:45:14

%S 1,12,60,90,-558,-2916,2160,61155,137619,-767880,-4940676,0,95128668,

%T 285386004,-974126979,-8413235910,-6504831279,142312459626,

%U 552074177142,-1081032363522,-13861905214518,-20792857821777,204246531941697,1012677253935633,-890531709052761

%N G.f. A(x) has the property that the first (n+1) terms of A(x)^(n+1) form the n-th row polynomial R_n(y) of triangle A097190 and satisfy R_n(1/3) = 9^n for all n>=0.

%H G. C. Greubel, <a href="/A097191/b097191.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: A(x) = 27*x/(1-(1-3*x)^9).

%F G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A097193.

%F a(18*n+11) = 0 for n>=0.

%e A(x) = 1 + 12*x + 60*x^2 + 90*x^3 - 558*x^4 - 2916*x^5 + 2160*x^6 +...

%e For n>=0, the first (n+1) coefficients of A(x)^(n+1) forms the

%e n-th row polynomial R_n(y) of triangle A097190:

%e A^1 = {1, _12,   60,    90,    -558,    -2916,       2160, ...}

%e A^2 = {1,  24, _264,  1620,    4644,    -8424,    -124524, ...}

%e A^3 = {1,  36,  612, _6318,   41526,   151956,     -16308, ...}

%e A^4 = {1,  48, 1104, 15912, _156744,  1061424,    4423032, ...}

%e A^5 = {1,  60, 1740, 32130,  417690, _3966732,   27243000, ...}

%e A^6 = {1,  72, 2520, 56700,  912492, 11027016, _101653164, ...}

%e These row polynomials satisfy: R_n(1/3) = 9^n:

%e 9^1 = 1 + 24/3;

%e 9^2 = 1 + 36/3 + 612/3^2;

%e 9^3 = 1 + 48/3 + 1104/3^2 + 15912/3^3;

%e 9^4 = 1 + 60/3 + 1740/3^2 + 32130/3^3 + 417690/3^4.

%p seq(coeff(series(27*x/(1-(1-3*x)^9), x, n+2), x, n), n = 0..30); # _G. C. Greubel_, Sep 17 2019

%t CoefficientList[Series[27*x/(1-(1-3*x)^9), {x,0,30}], x] (* _G. C. Greubel_, Sep 17 2019 *)

%o (PARI) a(n)=polcoeff(27*x/(1-(1-3*x)^9)+x*O(x^n),n,x)

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 27*x/(1-(1-3*x)^9) )); // _G. C. Greubel_, Sep 17 2019

%o (Sage)

%o def A097191_list(prec):

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

%o     return P(27*x/(1-(1-3*x)^9)).list()

%o A097191_list(30) # _G. C. Greubel_, Sep 17 2019

%Y Cf. A097190, A097193.

%K sign

%O 0,2

%A _Paul D. Hanna_, Aug 03 2004