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A323315 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 9*x*A(x) )^n * 4^n / 5^(n+1).

%I A323315 #7 Jan 10 2019 22:34:15
%S A323315 1,44,7096,1926724,711117536,325957899584,176862173366416,
%T A323315 110333447177205584,77614355506352291216,60715204091160869904064,
%U A323315 52262738608604757586146176,49089829530793158665498530304,49969859760169581295921965453056,54804935053668330661788935789639424,64439695005477056297527256416094395136,80874250846078911532650120181881418467904
%N A323315 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 9*x*A(x) )^n * 4^n / 5^(n+1).
%F A323315 G.f. A(x) satisfies the following identities.
%F A323315 (1) 1 = Sum_{n>=0} ( (1+x)^n - 9*x*A(x) )^n * 4^n / 5^(n+1).
%F A323315 (2) 1 = Sum_{n>=0} (1+x)^(n^2) * 4^n / (5 + 36*x*A(x)*(1+x)^n)^(n+1).
%e A323315 G.f.: A(x) = 1 + 44*x + 7096*x^2 + 1926724*x^3 + 711117536*x^4 + 325957899584*x^5 + 176862173366416*x^6 + 110333447177205584*x^7 + ...
%e A323315 such that
%e A323315 1 = 1/5 + ((1+x) - 9*x*A(x))*4/5^2 + ((1+x)^2 - 9*x*A(x))^2*4^2/5^3 + ((1+x)^3 - 9*x*A(x))^3*4^3/5^4 + ((1+x)^4 - 9*x*A(x))^4*4^4/5^5 + ...
%e A323315 Also,
%e A323315 1 = 1/(5 + 36*x*A(x)) + (1+x)*4/(5 + 36*x*A(x)*(1+x))^2 + (1+x)^4*4^2/(5 + 36*x*A(x)*(1+x)^2)^3 + (1+x)^9*4^3/(5 + 36*x*A(x)*(1+x)^3)^4 + ...
%o A323315 (PARI) \p120
%o A323315 {A=vector(1); A[1]=1; for(i=1,20, A = concat(A,0);
%o A323315 A[#A] = round( Vec( sum(n=0,1000, ( (1+x +x*O(x^#A))^n - 9*x*Ser(A) )^n * 4^n/5^(n+1)*1.)/36 ) )[#A+1]); A}
%Y A323315 Cf. A301435, A303288, A323314, A323316, A323317, A323318, A323319, A323320, A323321.
%K A323315 nonn
%O A323315 0,2
%A A323315 _Paul D. Hanna_, Jan 10 2019