A202438 G.f.: A(x) = ( Sum_{n>=0} 9^n*(2*n+1) * (-x)^(n*(n+1)/2) )^(-1/3).

1, 9, 162, 3537, 81405, 1944243, 47615121, 1186699005, 29960950842, 764012506770, 19637356382712, 507996422180784, 13211600995751697, 345145619340179829, 9051411187977957135, 238160821447956629934, 6284647075107225737511, 166263704846500625494533

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..17.

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.

Formula

a(5*n+4) == 0 (mod 5).

Self-convolution cube of A202437.

Conjectures: a(25*n+24) == 0 (mod 25) (checked up to n = 50) and a(7*n+5) == 0 (mod 7) (checked up to n = 200). - Peter Bala, Feb 26 2021

Example

G.f.: A(x) = 1 + 9*x + 162*x^2 + 3537*x^3 + 81405*x^4 + 1944243*x^5 +...
where
1/A(x)^3 = 1 - 27*x - 405*x^3 + 5103*x^6 + 59049*x^10 - 649539*x^15 - 6908733*x^21 +...+ 9^n*(2*n+1)*(-x)^(n*(n+1)/2) +...

Mathematica

nmax = 18;
Sum[9^n (2n+1)(-x)^(n(n+1)/2), {n, 0, nmax}]^(-1/3) + O[x]^nmax // CoefficientList[#, x]& (* Jean-François Alcover, Sep 09 2018 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, sqrtint(2*n+1), 9^m*(2*m+1)*(-x)^(m*(m+1)/2)+x*O(x^n))^(-1/3), n)}

Crossrefs

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

Sequence in context: A300843 A133681 A157553 * A237024 A156273 A051232

Adjacent sequences: A202435 A202436 A202437 * A202439 A202440 A202441

Keyword

nonn

Author

Paul D. Hanna, Dec 19 2011

Status

approved