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URL: https://oeis.org/A323315

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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).
9
1, 44, 7096, 1926724, 711117536, 325957899584, 176862173366416, 110333447177205584, 77614355506352291216, 60715204091160869904064, 52262738608604757586146176, 49089829530793158665498530304, 49969859760169581295921965453056, 54804935053668330661788935789639424, 64439695005477056297527256416094395136, 80874250846078911532650120181881418467904
OFFSET
0,2
FORMULA
G.f. A(x) satisfies the following identities.
(1) 1 = Sum_{n>=0} ( (1+x)^n - 9*x*A(x) )^n * 4^n / 5^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 4^n / (5 + 36*x*A(x)*(1+x)^n)^(n+1).
EXAMPLE
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 + ...
such that
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 + ...
Also,
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 + ...
PROG
(PARI) \p120
{A=vector(1); A[1]=1; for(i=1, 20, A = concat(A, 0);
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}
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 10 2019
STATUS
approved