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A128639
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Expansion of (1/3)* (c(q)^2/c(q^2))/ (b(q)^2/b(q^2)) in powers of q where b(), c() are cubic AGM analog functions.
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2
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1, 8, 40, 152, 488, 1392, 3640, 8896, 20584, 45512, 96816, 199200, 398072, 775216, 1475264, 2749776, 5029736, 9043344, 16005352, 27918304, 48047280, 81661504, 137183136, 227952960, 374924152, 610743224, 985891568, 1577869784
(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 (phi(-q^3)/ phi(-q))^4 in powers of q where phi() is a Ramanujan theta function.
Expansion of ((eta(q^3)/ eta(q))^2* (eta(q^2)/ eta(q^6)))^4 in powers of q.
Euler transform of period 6 sequence [ 8, 4, 0, 4, 8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u* (1-v)* (1-9*v) -(u-v)^2.
G.f.: (Product_{k>0} (1+x^k+x^(2k))/ (1-x^k+x^(2k)) )^4.
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EXAMPLE
| 1 + 8*q + 40*q^2 + 152*q^3 + 488*q^4 + 1392*q^5 + 3640*q^6 + ...
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PROG
| (PARI) {a(n) = local(A); if(n<0, 0, A = x*O(x^n); polcoeff( ((eta(x^3+A)/ eta(x+A))^2* eta(x^2+A)/ eta(x^6+A))^4, n))}
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CROSSREFS
| 8*A128638(n) = a(n) unless n = 0. Convolution inverse of A128637.
Sequence in context: A191903 A028596 A125198 * A004405 A001789 A074412
Adjacent sequences: A128636 A128637 A128638 * A128640 A128641 A128642
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Mar 16 2007
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