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A128640
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Expansion of (1/3)* (c(q^2)^2/c(q))/ (b(q^2)^2/b(q)) in powers of q where b(), c() are cubic AGM analog functions.
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4
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1, -4, 10, -20, 39, -76, 140, -244, 415, -696, 1140, -1820, 2861, -4448, 6816, -10292, 15372, -22756, 33356, -48408, 69683, -99600, 141312, -199036, 278557, -387608, 536230, -737632, 1009464, -1374888, 1863764, -2514868, 3378948, -4521672, 6027000, -8002676
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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* (psi(q^3)/ psi(q))^4 in powers of q where psi() is a Ramanujan theta function.
Expansion of ((eta(q^6)/ eta(q^2))^2* (eta(q)/ eta(q^3)))^4 in powers of q.
Euler transform of period 6 sequence [ -4, 4, 0, 4, -4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v* (1-u)* (1-9*u) -(u-v)^2.
G.f.: x* (Product_{k>0} (1-x^k+x^(2k))^2* (1+x^k+x^(2k)) )^4.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v)= (81*u^2*v^2 +9*u*v -12*u +30*u^2 -108*u^2*v +1)* v -u^3.
G.f. is a period 1 Fourier series which satisfies f(-1/ (6 t)) = (1/9) g(t) where q = exp(2 pi i t) and g() is g.f. for A128637.
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EXAMPLE
| q - 4*q^2 + 10*q^3 - 20*q^4 + 39*q^5 - 76*q^6 + 140*q^7 - 244*q^8 + ...
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PROG
| (PARI) {a(n) = local(A); if(n<1, 0, n--; A = x*O(x^n); polcoeff( ((eta(x^6+A)/ eta(x^2+A))^2* eta(x+A)/ eta(x^3+A))^4, n))}
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CROSSREFS
| A128641(n) = -a(n) unless n = 0. Convolution inverse of A128633.
Sequence in context: A090164 * A128641 A164617 A038421 A049032 A100354
Adjacent sequences: A128637 A128638 A128639 * A128641 A128642 A128643
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Mar 16 2007
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