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A323314 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 7*x*A(x) )^n * 3^n / 4^(n+1).

%I A323314 #7 Jan 10 2019 22:35:28
%S A323314 1,27,2625,429195,95328009,26290301175,8582072887881,3220902003386403,
%T A323314 1363088948866736193,641495666596787938899,332204944661961666375393,
%U A323314 187727027521862538450725607,114965661645391124805612197265,75859037026020765382177030210443,53662537374831689572836358288777665,40519124222573071898287923651933134187,32530810789422606721939134905409891249177,27674478227000422349878455201664033007066919
%N A323314 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 7*x*A(x) )^n * 3^n / 4^(n+1).
%F A323314 G.f. A(x) satisfies the following identities.
%F A323314 (1) 1 = Sum_{n>=0} ( (1+x)^n - 7*x*A(x) )^n * 3^n / 4^(n+1).
%F A323314 (2) 1 = Sum_{n>=0} (1+x)^(n^2) * 3^n / (4 + 21*x*A(x)*(1+x)^n)^(n+1).
%e A323314 G.f.: A(x) = 1 + 27*x + 2625*x^2 + 429195*x^3 + 95328009*x^4 + 26290301175*x^5 + 8582072887881*x^6 + 3220902003386403*x^7 + 1363088948866736193*x^8 + ...
%e A323314 such that
%e A323314 1 = 1/4 + ((1+x) - 7*x*A(x))*3/4^2 + ((1+x)^2 - 7*x*A(x))^2*3^2/4^3 + ((1+x)^3 - 7*x*A(x))^3*3^3/4^4 + ((1+x)^4 - 7*x*A(x))^4*3^4/4^5 + ...
%e A323314 Also,
%e A323314 1 = 1/(4 + 21*x*A(x)) + (1+x)*3/(4 + 21*x*A(x)*(1+x))^2 + (1+x)^4*3^2/(4 + 21*x*A(x)*(1+x)^2)^3 + (1+x)^9*3^3/(4 + 21*x*A(x)*(1+x)^3)^4 + ...
%o A323314 (PARI) \p120
%o A323314 {A=vector(1); A[1]=1; for(i=1,20, A = concat(A,0);
%o A323314 A[#A] = round( Vec( sum(n=0,1200, ( (1+x +x*O(x^#A))^n - 7*x*Ser(A) )^n * 3^n/4^(n+1)*1.)/21 ) )[#A+1]); A}
%Y A323314 Cf. A301435, A303288, A323315, A323316, A323317, A323318, A323319, A323320, A323321.
%K A323314 nonn
%O A323314 0,2
%A A323314 _Paul D. Hanna_, Jan 10 2019