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G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 13*x*A(x) )^n * 6^n / 7^(n+1).
9

%I #5 Jan 10 2019 22:14:31

%S 1,90,30360,17260998,13346871336,12819352461768,14575804541933076,

%T 19054882926950474988,28089490655708754330276,

%U 46046879475849529578435672,83060213421430745855381951856,163488644041366509740041070551248,348735916991281119541339971532867488,801490465035993025759896936239032263600,1974787497208210693752899355242321943894000,5193543503462268857667579481311302800804588450

%N G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 13*x*A(x) )^n * 6^n / 7^(n+1).

%F G.f. A(x) satisfies the following identities.

%F (1) 1 = Sum_{n>=0} ( (1+x)^n - 13*x*A(x) )^n * 6^n / 7^(n+1).

%F (2) 1 = Sum_{n>=0} (1+x)^(n^2) * 6^n / (7 + 78*x*A(x)*(1+x)^n)^(n+1).

%e G.f.: A(x) = 1 + 90*x + 30360*x^2 + 17260998*x^3 + 13346871336*x^4 + 12819352461768*x^5 + 14575804541933076*x^6 + 19054882926950474988*x^7 + ...

%e such that

%e 1 = 1/7 + ((1+x) - 13*x*A(x))*6/7^2 + ((1+x)^2 - 13*x*A(x))^2*6^2/7^3 + ((1+x)^3 - 13*x*A(x))^3*6^3/7^4 + ((1+x)^4 - 13*x*A(x))^4*6^4/7^5 + ...

%e Also,

%e 1 = 1/(7 + 78*x*A(x)) + (1+x)*6/(7 + 78*x*A(x)*(1+x))^2 + (1+x)^4*6^2/(7 + 78*x*A(x)*(1+x)^2)^3 + (1+x)^9*6^3/(7 + 78*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=0,1200, ( (1+x +x*O(x^#A))^n - 13*x*Ser(A) )^n * 6^n/7^(n+1)*1.)/78 ) )[#A+1]); A}

%Y Cf. A301435, A303288, A323314, A323315, A323316, A323318, A323319, A323320, A323321.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 10 2019