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Search: id:A139820
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| A139820 |
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Expansion of (phi(-q) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function. |
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1
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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, 137803776, -203554224
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OFFSET
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0,2
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
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FORMULA
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Expansion of (eta(q)^2 * eta(q^4) / eta(q^2)^3)^4 in powers of q.
Expansion of Jacobian elliptic function sqrt(k') in powers of nome q.
Euler transform of period 4 sequence [ -8, 4, -8, 0, ...].
G.f. A(x) satisifes 0 = f(A(x), A(x^2)) where f(u, v) = 4 * u - v^2 * (1 + u)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 4 g(t) where q = exp(2 pi i t) and g() is g.f. for A001938.
G.f.: ((Sum_{k} (-x)^k^2) / (Sum_{k} x^k^2))^2 = (Product_{k>0} (1 + x^(2*k)) / (1 + x^k)^2)^4.
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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 + A)^2 * eta(x^4 + A) / eta(x^2 + A)^3)^4, n))}
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CROSSREFS
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(-1)^n * A014969(n) = a(n). Convolution inverse of A014969.
Sequence in context: A053348 A019256 A014969 this_sequence A071345 A100312 A003201
Adjacent sequences: A139817 A139818 A139819 this_sequence A139821 A139822 A139823
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KEYWORD
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sign
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AUTHOR
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Michael Somos, May 01 2008
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