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A152243
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Expansion of a(q) * f(-q)^4 where f() is a Ramanujan theta function and a() is a cubic AGM function.
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1
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1, 2, -22, 26, 25, -46, 26, -22, -45, 0, 74, 122, -46, -142, 0, -44, 2, 194, -214, 0, 121, 146, 52, -22, 0, -286, -118, -262, 315, 50, 314, 0, -382, 386, 0, -166, -92, 338, 26, 0, -286, -572, 0, 52, 0, 242, 122, 458, 289, 0, -44, -358, -142, 0, -550, 362, 482, -188, -502, 0, 315, -718, 698, -694
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of q^(-1/2) * ( (eta(q) * eta(q^3))^3 + 3 * (eta(q^3) * eta(q^9))^3 ) in powers of q^3.
a(n) = b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) if p == 1 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 15552^(1/2) (t / i)^3 g(t) where q = exp(2 pi i t) and g() is g.f. for A152244.
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EXAMPLE
| q + 2*q^7 - 22*q^13 + 26*q^19 + 25*q^25 - 46*q^31 + 26*q^37 - 22*q^43 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, n *= 3; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^3 + 3 * x * (eta(x^3 + A) * eta(x^9 + A))^3, n))}
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CROSSREFS
| A152244(3*n) = A030208(3*n) = a(n).
Sequence in context: A111751 A037416 A057871 * A074160 A137074 A137106
Adjacent sequences: A152240 A152241 A152242 * A152244 A152245 A152246
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Nov 30 2008
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