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%I #24 Sep 13 2014 18:49:31
%S 1,1,22,1624,264962,79136637,38671111558,28642761340956,
%T 30413158977739302,44371247722932580948,86030010617294195269924,
%U 215785986252313362542154058,684854225048414942925120331598,2699657503162253569920254747627596,13010506207186236974375590663943378970
%N G.f. A(x) satisfies: A(x)^3 = Sum_{n>=0} A247031(n)*x^n.
%C The terms of A247031 satisfy:
%C 1 = Sum_{n>=0} A247031(n)*x^n * [Sum_{k=0..n+1} C(n+1, k)^3 * (-x)^k]^3.
%C Limit a(n)/A247031(n) = 1/3.
%C a(n) == 1 (mod 2) iff n = (4^k - 1)/3 for k>=0 (conjecture).
%C a(n) == 2 (mod 3) iff A247031(3*n) == 2 for n>=0 (mod 3) (conjecture), where A247031 is the self-convolution cube of this sequence.
%H Paul D. Hanna, <a href="/A245658/b245658.txt">Table of n, a(n) for n = 0..200</a>
%e G.f.: A(x) = 1 + x + 22*x^2 + 1624*x^3 + 264962*x^4 + 79136637*x^5 +...
%e where the coefficients of the cube of the g.f.
%e A(x)^3 = 1 + 3*x + 69*x^2 + 5005*x^3 + 806148*x^4 + 239220375*x^5 +...
%e satisfies:
%e 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 +...
%e OBSERVATIONS.
%e The terms of this sequence are odd at positions:
%e [0, 1, 5, 21, 85, 341, ..., (4^k-1)/3, ...].
%e The terms of this sequence are congruent to 2 modulo 3 at positions:
%e [4,12,31,36,37,38,40,58,85,93,108,109,110,111,114,120,166,174,247,255,...].
%e The terms of A247031 are congruent to 2 modulo 3 at positions:
%e [12,36,93,108,111,114,120,174,255,279,324,327,330,333,342,360,...].
%o (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))}
%o {a(n)=polcoeff(sum(m=0,n,A247031(m)*x^m +x*O(x^n))^(1/3), n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A247031.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 13 2014