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A097181 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 8^n, where R_n(y) forms the initial (n+1) terms of g.f. A097182(y)^(n+1).

%I A097181 #11 Sep 17 2019 21:22:13
%S A097181 1,1,14,1,21,210,1,28,378,3220,1,35,595,6475,49910,1,42,861,11396,
%T A097181 108402,778596,1,49,1176,18326,207074,1791930,12198004,1,56,1540,
%U A097181 27608,361018,3647672,29389492,191682920,1,63,1953,39585,587727,6783147
%N A097181 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 8^n, where R_n(y) forms the initial (n+1) terms of g.f. A097182(y)^(n+1).
%C A097181 Row sums form A097185. Diagonal is A097183.
%C A097181 Ratio of g.f.s of any two adjacent diagonals equals g.f. of A097184, where the g.f.s satisfy: A097182(x*A097184(x)) = A097184(x).
%H A097181 G. C. Greubel, <a href="/A097181/b097181.txt">Rows n = 0..50 of triangle, flattened</a>
%F A097181 G.f.: A(x, y) = 2*y/((1-16*x*y) + (2*y-1)*(1-16*x*y)^(7/8)).
%F A097181 G.f.: A(x, y) = A097183(x*y)/(1 - x*A097184(x*y)).
%e A097181 Row polynomials evaluated at y=1/2 equals powers of 8:
%e A097181 8^1 = 1 + 14/2;
%e A097181 8^2 = 1 + 21/2 + 210/2^2;
%e A097181 8^3 = 1 + 28/2 + 378/2^2 + 3220/2^3;
%e A097181 8^4 = 1 + 35/2 + 595/2^2 + 6475/2^3 + 49910/2^4;
%e A097181 where A097182(y)^(n+1) has the same initial terms as the n-th row:
%e A097181 A097182(y) = 1 + 7*x + 21*x^2 + 21*x^3 - 63*x^4 - 231*x^5 -+...
%e A097181 A097182(y)^2 = 1 + 14y +...
%e A097181 A097182(y)^3 = 1 + 21y + 210y^2 +...
%e A097181 A097182(y)^4 = 1 + 28y + 378y^2 + 3220y^3 +...
%e A097181 A097182(y)^5 = 1 + 35y + 595y^2 + 6475y^3 + 49910y^4 +...
%e A097181 Rows begin with n=0:
%e A097181   1;
%e A097181   1, 14;
%e A097181   1, 21,  210;
%e A097181   1, 28,  378,  3220;
%e A097181   1, 35,  595,  6475,  49910;
%e A097181   1, 42,  861, 11396, 108402,  778596;
%e A097181   1, 49, 1176, 18326, 207074, 1791930, 12198004;
%e A097181   1, 56, 1540, 27608, 361018, 3647672, 29389492, 191682920;
%e A097181   1, 63, 1953, 39585, 587727, 6783147, 62974371, 479497491, 3019005990; ...
%t A097181 Table[SeriesCoefficient[2*y/((1-16*x*y) + (2*y-1)*(1-16*x*y)^(7/8)), {x, 0,n}, {y,0,k}], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 17 2019 *)
%o A097181 (PARI) {T(n,k)=if(n==0,1,if(k==0,1,if(k==n, 2^n*(4^n -sum(j=0,n-1, T(n,j)/2^j)), polcoeff((Ser(vector(n,i,T(n-1,i-1)),x) +x*O(x^k))^((n+1)/n),k,x))))}
%Y A097181 Cf. A097179, A097182, A097183, A097184, A097185, A097186.
%K A097181 nonn,tabl
%O A097181 0,3
%A A097181 _Paul D. Hanna_, Aug 03 2004