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A323317
G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 13*x*A(x) )^n * 6^n / 7^(n+1).
9
1, 90, 30360, 17260998, 13346871336, 12819352461768, 14575804541933076, 19054882926950474988, 28089490655708754330276, 46046879475849529578435672, 83060213421430745855381951856, 163488644041366509740041070551248, 348735916991281119541339971532867488, 801490465035993025759896936239032263600, 1974787497208210693752899355242321943894000, 5193543503462268857667579481311302800804588450
OFFSET
0,2
FORMULA
G.f. A(x) satisfies the following identities.
(1) 1 = Sum_{n>=0} ( (1+x)^n - 13*x*A(x) )^n * 6^n / 7^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 6^n / (7 + 78*x*A(x)*(1+x)^n)^(n+1).
EXAMPLE
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 + ...
such that
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 + ...
Also,
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 + ...
PROG
(PARI) \p120
{A=vector(1); A[1]=1; for(i=1, 20, A = concat(A, 0);
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}
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 10 2019
STATUS
approved