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A111983
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G.f.: A(x) = Sum_{n>=0} (2*n+1) * 8^n * x^(n*(n+1)/2).
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5
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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).
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.
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REFERENCES
| G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 285.
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LINKS
| 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.
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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 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 +-...
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MAPLE
| add((2*n+1) * 8^n * x^(n*(n+1)/2), n=0..50);
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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)
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CROSSREFS
| Cf. A111984 (g.f. A(x)^(1/3)), A111985 (g.f. A(x)^(1/12)).
Sequence in context: A075404 A194894 A128379 * A174560 A040581 A040582
Adjacent sequences: A111980 A111981 A111982 * A111984 A111985 A111986
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 25 2005
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