A111983 G.f.: A(x) = Sum_{n>=0} (2*n+1) * 8^n * x^(n*(n+1)/2).

1, 24, 0, 320, 0, 0, 3584, 0, 0, 0, 36864, 0, 0, 0, 0, 360448, 0, 0, 0, 0, 0, 3407872, 0, 0, 0, 0, 0, 0, 31457280, 0, 0, 0, 0, 0, 0, 0, 285212672, 0, 0, 0, 0, 0, 0, 0, 0, 2550136832, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22548578304, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 197568495616, 0, 0, 0, 0, 0, 0

Offset

0,2

Comments

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).

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. - Eric Rains and N. J. A. Sloane, Nov 06 2005

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..5050

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

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 section), 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 +- ...

Maple

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

Mathematica

CoefficientList[Sum[(2n+1) 8^n x^(n(n+1)/2), {n, 0, 12}], x] (* Jean-François Alcover, Jul 26 2018 *)

Prog

(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)
(Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(2*n+1)*8^n*x^Binomial(n+1, 2): n in [0..m]]) )); // G. C. Greubel, Oct 24 2019
(Sage)
def A052961_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( sum((2*n+1)*8^n*x^binomial(n+1, 2) for n in (0..13))).list()
A052961_list(80) # G. C. Greubel, Oct 24 2019

Crossrefs

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

Sequence in context: A194894 A364226 A128379 * A326857 A340135 A364203

Adjacent sequences: A111980 A111981 A111982 * A111984 A111985 A111986

Keyword

nonn

Author

Paul D. Hanna, Aug 25 2005

Status

approved