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A014972
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Expansion of (theta_3 / theta_4)^4; also of 1/(1-lambda(z)).
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3
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1, 16, 128, 704, 3072, 11488, 38400, 117632, 335872, 904784, 2320128, 5702208, 13504512, 30952544, 68901888, 149403264, 316342272, 655445792, 1331327616, 2655115712, 5206288384, 10049485312, 19115905536, 35867019904, 66437873664
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The relation with A092877 is equivalent to eta(q^2)^24=eta(q)^16*et(q^4)^8+16*eta(q)^8*eta(q^4)^16. - Michael Somos Apr 11 2004
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REFERENCES
| J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
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LINKS
| Eric Weisstein's World of Mathematics, Elliptic Lambda Function
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FORMULA
| G.f.: (Product_{k>0} (1+x^(2k-1))/(1-x^(2k-1)))^8 = exp(16*Sum_{k>0} x^(2k-1)sigma(2k-1)/(2k-1)). - Michael Somos Apr 11 2004
Euler transform of period 4 sequence [ 16, -8, 16, 0, ...]. - Michael Somos Apr 11 2004
Expansion of (eta(q^2)^3/(eta(q^4)eta(q)^2))^8 in powers of q. - Michael Somos Apr 11 2004
G.f. A(x) satisfies A(-x)=1/A(x). Also 0=f(A(x), A(x^2)) where f(u, v)= (u-1)^2+16uv(1-v). - Michael Somos Apr 11 2004
Empirical : sum(exp(-2*Pi)^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = -16+12*2^(1/2). Simon Plouffe, Feb. 20, 2011.
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PROG
| (PARI) a(n)=if(n<0, 0, polcoeff(exp(16*sum(k=1, (n+1)\2, sigma(2*k-1)/(2*k-1)*x^(2*k-1), x*O(x^n))), n)) /* Michael Somos Apr 11 2004 */
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff((eta(x^2+A)^3/eta(x+A)^2/eta(x^4+A))^8, n))} /* Michael Somos Apr 11 2004 */
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CROSSREFS
| Cf. A005798, A007248, A029845.
a(n)=16*A092877(n), if n>0.
Sequence in context: * A115977 A128692 A132136 A163399 A067488 A120785
Adjacent sequences: A014969 A014970 A014971 * A014973 A014974 A014975
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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