0,2

Define F(x,q) = Sum_{n>=0} q^n*(2*n+1)*x^(n*(n+1)/2). (1) F(x,q)^(1/3) is an integer series in x when q == -1, 0, 3 or 6 (mod 9). (2) For q = -1 we have the famous result of Jacobi (Hardy and Wright, Th. 357): F(x,-1)^(1/3) = (1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 +...)^(1/3) = 1 + Sum_{n>=1} (-1)^n*[x^(n*(3*n-1)/2)+x^(n*(3*n+1)/2)] = Product_{k>=1} (1-x^k).

Comments from Eric Rains and N. J. A. Sloane (njas(AT)research.att.com), Nov 06 2005: Concerning (1): For q == 0 mod 3 we see that F == 1 mod 9, which by the Heninger-Rains-Sloane paper implies that F^(1/3) has integer coefficients. For q == -1 mod 9 the same assertion follows from the Jacobi identity mentioned above. For q = 8, F(x,8) = A(x), the current sequence, we see that A == 1 mod 8, so A^(1/3) == 1 mod 8 and then, again by our paper, A^(1/12) has integer coefficients.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 285.

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.

A(x) = 1 + 3*8*x + 5*8^2*x^3 + 7*8^3*x^6 + 9*8^4*x^10 +... = 1 + 24*x + 320*x^3 + 3584*x^6 + 36864*x^10 + 360448*x^15 + 3407872*x^21 + 31457280*x^28 + 285212672*x^36 + 2550136832*x^45 + ...

Surprisingly, A(x)^(1/3) is an integer series (A111984):

A(x)^(1/3) = 1 + 8*x - 64*x^2 + 960*x^3 - 15360*x^4 +-...

In fact (see proof in Comments line), A(x)^(1/12) is also an integer series (A111985):

A(x)^(1/12) = 1 + 2*x - 22*x^2 + 364*x^3 - 6490*x^4 +-...

add((2*n+1) * 8^n * x^(n*(n+1)/2), n=0..50);

(PARI) a(n)=polcoeff(sum(k=0, sqrtint(2*n+1), (2*k+1)*8^k*x^(k*(k+1)/2)+x*O(x^n)), n)

Cf. A111984 (g.f. A(x)^(1/3)), A111985 (g.f. A(x)^(1/12)).

Sequence in context: A140793 A023923 A128379 this_sequence A040581 A040582 A102914

Adjacent sequences: A111980 A111981 A111982 this_sequence A111984 A111985 A111986

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 25 2005