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A097179 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 4^n, where R_n(y) forms the initial (n+1) terms of g.f. A077860(y)^(n+1).

%I A097179 #7 Sep 17 2019 22:24:27
%S A097179 1,1,6,1,9,42,1,12,74,308,1,15,115,595,2310,1,18,165,1020,4746,17556,
%T A097179 1,21,224,1610,8722,37730,134596,1,24,292,2392,14778,73080,299508,
%U A097179 1038312,1,27,369,3393,23535,130851,604707,2376099,8046918
%N A097179 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 4^n, where R_n(y) forms the initial (n+1) terms of g.f. A077860(y)^(n+1).
%C A097179 Row sums form A097180. Diagonal is A004982. Ratio of g.f.s of any two adjacent diagonals equals g.f. of A048779, where the g.f.s satisfy: A077860(x*A048779(x)) = A048779(x).
%H A097179 G. C. Greubel, <a href="/A097179/b097179.txt">Rows n = 0..50 of triangle, flattened</a>
%F A097179 G.f.: A(x, y) = 2*y/((1-8*x*y) + (2*y-1)*(1-8*x*y)^(3/4)).
%F A097179 G.f.: A(x, y) = A004982(x*y)/(1 - x*A048779(x*y)).
%e A097179 Row polynomials evaluated at y=1/2 equals powers of 4:
%e A097179 4^1 = 1 + 6/2;
%e A097179 4^2 = 1 + 9/2 + 42/2^2;
%e A097179 4^3 = 1 + 12/2 + 74/2^2 + 308/2^3;
%e A097179 4^4 = 1 + 15/2 + 115/2^2 + 595/2^3 + 2310/2^4;
%e A097179 where A077860(y)^(n+1) has the same initial terms as the n-th row:
%e A097179 A077860(y) = 1 +3*y +5*y^2 +5*y^3 +1*y^4 -7*y^5 -15*y^6 -15*y^7 +...
%e A097179 A077860(y)^2 = 1 + 6*y +...
%e A097179 A077860(y)^3 = 1 + 9*y + 42*y^2 +...
%e A097179 A077860(y)^4 = 1 + 12*y + 74*y^2 + 308*y^3 +...
%e A097179 A077860(y)^5 = 1 + 15*y + 115*y^2 + 595*y^3 + 2310*y^4 +...
%e A097179 Rows begin with n=0:
%e A097179   1;
%e A097179   1,  6;
%e A097179   1,  9,  42;
%e A097179   1, 12,  74,  308;
%e A097179   1, 15, 115,  595,  2310;
%e A097179   1, 18, 165, 1020,  4746, 17556;
%e A097179   1, 21, 224, 1610,  8722, 37730,  134596;
%e A097179   1, 24, 292, 2392, 14778, 73080,  299508, 1038312;
%e A097179   1, 27, 369, 3393, 23535, 130851, 604707, 2376099, 8046918; ...
%t A097179 Table[SeriesCoefficient[2*y/((1-8*x*y) +(2*y-1)*(1-8*x*y)^(3/4)), {x, 0, n}, {y,0,k}], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 17 2019 *)
%o A097179 (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 A097179 Cf. A004982, A048779, A077860, A097180, A097181.
%K A097179 nonn,tabl
%O A097179 0,3
%A A097179 _Paul D. Hanna_, Aug 02 2004