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Search: id:A014969
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| A014969 |
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Expansion of (theta_3 / theta_4)^2. |
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5
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| 1, 8, 32, 96, 256, 624, 1408, 3008, 6144, 12072, 22976, 42528, 76800, 135728, 235264, 400704, 671744, 1109904, 1809568, 2914272, 4640256, 7310592, 11404416, 17626944, 27009024, 41047992, 61905088, 92681664
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OFFSET
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0,2
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 375. Eq. (17)
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
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FORMULA
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Euler transform of period 4 sequence [8, -4, 8, 0, ...]. - Michael Somos, Jul 07 2005
G.f.: (theta_3/theta_4)^2 = (Sum_{k} x^k^2)/(Sum_{k} (-x)^k^2)^2 = (Product_{k>0} (1-x^(4k-2))/((1-x^(4k-1))(1-x^(4k-3)))^2)^4.
Expansion of Fricke tau_8(omega)/2+1 in powers of q = exp(2*pi*i*z).
Expansion of elliptic 1/sqrt(1-lambda(z))=1/k' in powers of nome q = exp(pi*i*z).
G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v) = (1+u)^2 -4*u*v^2 . - Michael Somos Nov 14 2006
Expansion of (phi(q) / phi(-q))^2 in powers of q where phi() is a Ramanujan theta function.
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EXAMPLE
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1 + 8*q + 32*q^2 + 96*q^3 + 256*q^4 + 624*q^5 + 1408*q^6 + 3008*q^7 + ...
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PROGRAM
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(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))^4, n))} /* Michael Somos Jul 07 2005 */
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CROSSREFS
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8 * A107035(n) = a(n) unless n=0. 2 * A131126(n) = a(n) unless n=0.
Sequence in context: A159941 A053348 A019256 this_sequence A139820 A071345 A100312
Adjacent sequences: A014966 A014967 A014968 this_sequence A014970 A014971 A014972
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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