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A128642
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Expansion of (b(q)/ b(q^2))^3 in powers of q where b() is a cubic AGM analog function.
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2
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1, -9, 36, -90, 180, -351, 684, -1260, 2196, -3735, 6264, -10260, 16380, -25749, 40032, -61344, 92628, -138348, 204804, -300204, 435672, -627147, 896400, -1271808, 1791324, -2507013, 3488472, -4826070, 6638688, -9085176, 12373992, -16773876, 22633812
(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 (chi(-q)^3/ chi(-q^2))^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of ((eta(q)/ eta(q^2))^3* (eta(q^6)/ eta(q^3)))^3 in powers of q.
Euler transform of period 6 sequence [ -9, 0, -6, 0, -9, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u* (1-v)* (8+u) -(u-v)^2.
G.f.: (Product_{k>0} (1+x^(3k))/ (1+x^k)^3)^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 8 g(t) where q = exp(2 pi i t) and g() is g.f. for A123633.
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EXAMPLE
| 1 - 9*q + 36*q^2 - 90*q^3 + 180*q^4 - 351*q^5 + 684*q^6 - 1260*q^7 + ...
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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^2+A))^3* eta(x^6+A)/ eta(x^3+A))^3, n))}
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CROSSREFS
| 9*A128640(n) = -a(n) unless n = 0. Convolution inverse of A128643.
Sequence in context: A045851 A173243 A162258 * A022604 A085630 A133226
Adjacent sequences: A128639 A128640 A128641 * A128643 A128644 A128645
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Mar 16 2007
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