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A128641
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Expansion of (1/3)* (c(q)^2/c(q^2))/ (b(q^2)^2/b(q)) in powers of q where b(), c() are cubic AGM analog functions.
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3
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1, -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
| 0,3
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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)^3/ phi(-q))/ (psi(q)^3/ psi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q)/ eta(q^6))* (eta(q^3)/ eta(q^2))^5 in powers of q.
Euler transform of period 6 sequence [ -1, 4, -6, 4, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u* (1-v)* (8-9*u) +(u-v)^2.
G.f.: Product_{k>0} (1-x^k)/ (1-x^(6k))* ((1-(x^3k))/ (1-x^(2k)))^5.
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EXAMPLE
| 1 - q + 4*q^2 - 10*q^3 + 20*q^4 - 39*q^5 + 76*q^6 - 140*q^7 + ...
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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^2+A))^5* eta(x+A)/ eta(x^6+A), n))}
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CROSSREFS
| A128640(n) = -a(n) unless n = 0. Convolution inverse of A128636.
Sequence in context: A090164 A128640 * A164617 A038421 A049032 A100354
Adjacent sequences: A128638 A128639 A128640 * A128642 A128643 A128644
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Mar 16 2007
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