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%I #5 Jan 10 2019 22:19:29
%S 1,152,87760,85439240,113151839104,186152435786240,362548564958149696,
%T 811847325733606058048,2049967057729258844550208,
%U 5756221555712461523954507264,17785396518936498493080842349568,59963943179536216027803213130483712,219093913413498532617018883655015864320,862506026576114820987041351988191302565888,3640101913203153345185251232178995247004487680,16397805545827151302219567488776238270687543337472
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 17*x*A(x) )^n * 8^n / 9^(n+1).
%F G.f. A(x) satisfies the following identities.
%F (1) 1 = Sum_{n>=0} ( (1+x)^n - 17*x*A(x) )^n * 8^n / 9^(n+1).
%F (2) 1 = Sum_{n>=0} (1+x)^(n^2) * 8^n / (9 + 136*x*A(x)*(1+x)^n)^(n+1).
%e G.f.: A(x) = 1 + 152*x + 87760*x^2 + 85439240*x^3 + 113151839104*x^4 + 186152435786240*x^5 + 362548564958149696*x^6 + 811847325733606058048*x^7 + ...
%e such that
%e 1 = 1/9 + ((1+x) - 17*x*A(x))*8/9^2 + ((1+x)^2 - 17*x*A(x))^2*8^2/9^3 + ((1+x)^3 - 17*x*A(x))^3*8^3/9^4 + ((1+x)^4 - 17*x*A(x))^4*8^4/9^5 + ...
%e Also,
%e 1 = 1/(9 + 136*x*A(x)) + (1+x)*8/(9 + 136*x*A(x)*(1+x))^2 + (1+x)^4*8^2/(9 + 136*x*A(x)*(1+x)^2)^3 + (1+x)^9*8^3/(9 + 136*x*A(x)*(1+x)^3)^4 + ...
%o (PARI) \p120
%o {A=vector(1); A[1]=1; for(i=1,20, A = concat(A,0);
%o A[#A] = round( Vec( sum(n=0, 2200, ( (1+x +x*O(x^#A))^n - 17*x*Ser(A) )^n * 8^n/9^(n+1)*1.)/136 ) )[#A+1]); A}
%Y Cf. A301435, A303288, A323314, A323315, A323316, A323317, A323318, A323320, A323321.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 10 2019
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