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EXAMPLE
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G.f.: A(x) = 1 + x + 22*x^2 + 1624*x^3 + 264962*x^4 + 79136637*x^5 +...
where the coefficients of the cube of the g.f.
A(x)^3 = 1 + 3*x + 69*x^2 + 5005*x^3 + 806148*x^4 + 239220375*x^5 +...
satisfies:
1 = 1*(1-x)^3 + 3*x*(1-2^3*x+x^2)^3 + 69*x^2*(1-3^3*x+3^3*x^2-x^3)^3 + 5005*x^3*(1-4^3*x+6^3*x^2-4^3*x^3+x^4)^3 + 806148*x^4*(1-5^3*x+10^3*x^2-10^3*x^3+5^3*x^4-x^5)^3 +...
OBSERVATIONS.
The terms of this sequence are odd at positions:
[0, 1, 5, 21, 85, 341, ..., (4^k-1)/3, ...].
The terms of this sequence are congruent to 2 modulo 3 at positions:
[4,12,31,36,37,38,40,58,85,93,108,109,110,111,114,120,166,174,247,255,...].
The terms of A247031 are congruent to 2 modulo 3 at positions:
[12,36,93,108,111,114,120,174,255,279,324,327,330,333,342,360,...].
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PROG
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(PARI) {A247031(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, A247031(m)*x^m*sum(k=0, m+1, binomial(m+1, k)^3 * (-x)^k +x*O(x^n) )^3 ), n))}
{a(n)=polcoeff(sum(m=0, n, A247031(m)*x^m +x*O(x^n))^(1/3), n)}
for(n=0, 20, print1(a(n), ", "))
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