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A202436
G.f.: A(x) = ( Sum_{n>=0} (-10)^n*(2*n+1) * x^(n*(n+1)/2) )^(-1/3).
1
1, 10, 200, 4500, 110000, 2800000, 73169000, 1946760000, 52486600000, 1429524000000, 39248429970000, 1084632798800000, 30135969080000000, 841120372160000000, 23567430432900000000, 662548090558333700000, 18680473491148068000000
OFFSET
0,2
COMMENTS
Compare to the q-series identity:
1/P(x)^3 = Sum_{n>=0} (-1)^n*(2*n+1) * x^(n*(n+1)/2),
where P(x) is the partition function (g.f. of A000041).
LINKS
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
EXAMPLE
G.f.: A(x) = 1 + 10*x + 200*x^2 + 4500*x^3 + 110000*x^4 + 2800000*x^5 +...
where
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) +...
MATHEMATICA
nmax = 17;
A[x_] = Sum[(-10)^n (2n+1) x^(n(n+1)/2), {n, 0, nmax}]^(-1/3) + O[x]^nmax;
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 27 2018 *)
PROG
(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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 19 2011
STATUS
approved