A202437 G.f.: A(x) = ( Sum_{n>=0} 9^n*(2*n+1) * (-x)^(n*(n+1)/2) )^(-1/9).

1, 3, 45, 900, 19305, 437076, 10254681, 246553875, 6035226975, 149777902710, 3757716928053, 95110270281675, 2424907723685985, 62204709603345075, 1604054030028748830, 41549974064592136020, 1080505644116115671622, 28195636842752845510215, 738014045325584817820275

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..18.

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+2) == a(5*n+3) == a(5*n+4) == 0 (mod 5).

Self-convolution cube yields A202438.

Example

G.f.: A(x) = 1 + 3*x + 45*x^2 + 900*x^3 + 19305*x^4 + 437076*x^5 +...
where
1/A(x)^9 = 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) +...
Note that the residues a(n) (mod 5) begin:
[1,3,0,0,0,1,1,0,0,0,3,0,0,0,0,0,2,0,0,0,2,2,0,0,0,1,3,0,0,0,3,3,0,0,0,4,4...].

Mathematica

nmax = 19;
Sum[9^n (2n+1)(-x)^(n(n+1)/2), {n, 0, nmax}]^(-1/9) + 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))^(-1/9), n)}

Crossrefs

Cf. A202210, A202436, A202438, A193236, A193237, A111984.

Sequence in context: A266698 A132303 A298799 * A008931 A036278 A225149

Adjacent sequences: A202434 A202435 A202436 * A202438 A202439 A202440

Keyword

nonn

Author

Paul D. Hanna, Dec 19 2011

Status

approved