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A131946
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Expansion of (eta(q) * eta(q^3))^4/( eta(q^2) * eta(q^6))^2 in powers of q.
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2
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1, -4, 4, -4, 20, -24, 4, -32, 52, -4, 24, -48, 20, -56, 32, -24, 116, -72, 4, -80, 120, -32, 48, -96, 52, -124, 56, -4, 160, -120, 24, -128, 244, -48, 72, -192, 20, -152, 80, -56, 312, -168, 32, -176, 240, -24, 96, -192, 116, -228, 124, -72, 280, -216, 4, -288, 416, -80, 120, -240, 120, -248, 128, -32
(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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REFERENCES
| N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.66).
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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 (phi(-q)* phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function.
Expansion of (4*a(q^2)^2 -a(q)^2)/3 = (b(q)^2/b(q^2))* (c(q)^2/c(q^2))/3 in powers of q where a(),b(),c() are cubic AGM function.
Euler transform of period 6 sequence [ -4, -2, -8, -2, -4, -4, ...].
G.f.: 1 -4*( Sum_{k>0} k* (-x)^k/(1-x^k)* kronecker(9, k)) = (theta_3(-x)* theta_3(-x^3))^2.
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PROG
| (PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( (eta(x+A)* eta(x^3+A))^4/( eta(x^2+A)* eta(x^6+A))^2, n))}
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CROSSREFS
| (-1)^n*A034896(n) = a(n). -4*A131947(n) = a(n) unless n = 0.
Sequence in context: A141666 A102127 A201625 * A034896 A120914 A024949
Adjacent sequences: A131943 A131944 A131945 * A131947 A131948 A131949
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Jul 30 2007
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