A202436 G.f.: A(x) = ( Sum_{n>=0} (-10)^n*(2*n+1) * x^(n*(n+1)/2) )^(-1/3).

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

Table of n, a(n) for n=0..16.

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

Cf. A111984, A202437, A202438, A193236, A193237, A202210.

Sequence in context: A354410 A126431 A335649 * A320671 A237025 A156275

Adjacent sequences: A202433 A202434 A202435 * A202437 A202438 A202439

Keyword

nonn

Author

Paul D. Hanna, Dec 19 2011

Status

approved