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A097186 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/3) = 3^n, where R_n(y) forms the initial (n+1) terms of g.f. A057083(y)^(n+1).

%I A097186 #8 Sep 17 2019 15:28:37
%S A097186 1,1,6,1,9,45,1,12,78,360,1,15,120,675,2970,1,18,171,1134,5859,24948,
%T A097186 1,21,231,1764,10458,51030,212058,1,24,300,2592,17334,95256,445824,
%U A097186 1817640,1,27,378,3645,27135,165726,861597,3905253,15677145,1,30,465,4950,40590,272646,1557765,7760610,34285680,135868590
%N A097186 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/3) = 3^n, where R_n(y) forms the initial (n+1) terms of g.f. A057083(y)^(n+1).
%C A097186 Row sums form A097189. Main diagonal is A004988. Ratio of g.f.s of any two adjacent diagonals equals g.f. of A097188, where the g.f.s satisfy: A057083(x*A097188(x)) = A097188(x).
%H A097186 G. C. Greubel, <a href="/A097186/b097186.txt">Rows n = 0..50 of triangle, flattened</a>
%F A097186 G.f.: A(x, y) = 3*y/((1-9*x*y) + (3*y-1)*(1-9*x*y)^(2/3)).
%F A097186 G.f.: A(x, y) = A004988(x*y)/(1 - x*A097188(x*y)).
%e A097186 Row polynomials evaluated at y=1/3 equals powers of 3:
%e A097186 3^1 = 1 + 6/3;
%e A097186 3^2 = 1 + 9/3 + 45/3^2;
%e A097186 3^3 = 1 + 12/3 + 78/3^2 + 360/3^3;
%e A097186 3^4 = 1 + 15/3 + 120/3^2 + 675/3^3 + 2970/3^4;
%e A097186 where A057083(y)^(n+1) has the same initial terms as the n-th row:
%e A097186 A057083(y) = 1 + 3y + 6y^2 + 9y^3 + 9y^4 + 0y^5 - 27y^6 +...
%e A097186 A057083(y)^2 = 1 + 6y +...
%e A097186 A057083(y)^3 = 1 + 9y + 45y^2 +...
%e A097186 A057083(y)^4 = 1 + 12y + 78y^2 + 360y^3 +...
%e A097186 A057083(y)^5 = 1 + 15y + 120y^2 + 675y^3 + 2970y^4 +...
%e A097186 Rows begin with n=0:
%e A097186   1;
%e A097186   1,  6;
%e A097186   1,  9,  45;
%e A097186   1, 12,  78,  360;
%e A097186   1, 15, 120,  675,  2970;
%e A097186   1, 18, 171, 1134,  5859,  24948;
%e A097186   1, 21, 231, 1764, 10458,  51030, 212058;
%e A097186   1, 24, 300, 2592, 17334,  95256, 445824, 1817640;
%e A097186   1, 27, 378, 3645, 27135, 165726, 861597, 3905253, 15677145; ...
%t A097186 Table[SeriesCoefficient[3y/((1-9xy) - (1-3y)*(1-9xy)^(2/3)), {x,0,n}, {y,0,k}], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 17 2019 *)
%o A097186 (PARI) {T(n,k)=if(n==0,1,if(k==0,1,if(k==n, 3^n*(3^n -sum(j=0,n-1, T(n,j)/3^j)), polcoeff((Ser(vector(n,i,T(n-1,i-1)), x) +x*O(x^k))^((n+1)/n),k,x))))}
%Y A097186 Cf. A004988, A057083, A097188, A097189, A097190.
%K A097186 nonn,tabl
%O A097186 0,3
%A A097186 _Paul D. Hanna_, Aug 03 2004
%E A097186 More terms added by _G. C. Greubel_, Sep 17 2019