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%I #13 Jul 27 2018 09:37:17
%S 1,10,200,4500,110000,2800000,73169000,1946760000,52486600000,
%T 1429524000000,39248429970000,1084632798800000,30135969080000000,
%U 841120372160000000,23567430432900000000,662548090558333700000,18680473491148068000000
%N G.f.: A(x) = ( Sum_{n>=0} (-10)^n*(2*n+1) * x^(n*(n+1)/2) )^(-1/3).
%C Compare to the q-series identity:
%C 1/P(x)^3 = Sum_{n>=0} (-1)^n*(2*n+1) * x^(n*(n+1)/2),
%C where P(x) is the partition function (g.f. of A000041).
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006.
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://doi.org/10.1016/j.jcta.2006.03.018">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%e G.f.: A(x) = 1 + 10*x + 200*x^2 + 4500*x^3 + 110000*x^4 + 2800000*x^5 +...
%e where
%e 1/A(x)^3 = 1 - 30*x + 500*x^3 - 7000*x^6 + 90000*x^10 - 1100000*x^15 +...+ (-10)^n*(2*n+1)*x^(n*(n+1)/2) +...
%t nmax = 17;
%t A[x_] = Sum[(-10)^n (2n+1) x^(n(n+1)/2), {n, 0, nmax}]^(-1/3) + O[x]^nmax;
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Jul 27 2018 *)
%o (PARI) {a(n)=local(S=sum(m=0, sqrtint(2*n), (-10)^m*(2*m+1)*x^(m*(m+1)/2))+x*O(x^n)); polcoeff(S^(-1/3), n)}
%Y Cf. A111984, A202437, A202438, A193236, A193237, A202210.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 19 2011
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