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Search: id:A131943
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| A131943 |
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Expansion of eta(q)^3 * eta(q^2)^3/( eta(q^3) * eta(q^6)) in powers of q. |
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3
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| 1, -3, -3, 15, -3, -18, 15, -24, -3, 69, -18, -36, 15, -42, -24, 90, -3, -54, 69, -60, -18, 120, -36, -72, 15, -93, -42, 231, -24, -90, 90, -96, -3, 180, -54, -144, 69, -114, -60, 210, -18, -126, 120, -132, -36, 414, -72, -144, 15, -171, -93, 270, -42, -162, 231, -216, -24, 300, -90, -180, 90, -186, -96, 552, -3, -252
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.65).
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FORMULA
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Expansion of b(q)*b(q^2) where b() is a cubic AGM function.
Euler transform of period 6 sequence [ -3, -6, -2, -6, -3, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1/ (6 t)) = 54 (t/i)^2 g(t) where q = exp(2 pi i t) and g() is g.f. for A121443.
G.f.: Product_{k>0} ((1-x^k)* (1-x^(2k)))^3/( (1-x^(3k))* (1-x^(6k))).
G.f.: 1 -3*(Sum_{k>0} (6k-1)* x^(6k-1)/( 1-x^(6k-1)) -2*(6k-5)* x^(6k-3)/( 1-x^(6k-3)) +(6k-5)* x^(6k-5)/( 1-x^(6k-5))).
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EXAMPLE
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1 - 3*q - 3*q^2 + 15*q^3 - 3*q^4 - 18*q^5 + 15*q^6 - 24*q^7 - 3*q^8 +...
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PROGRAM
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(PARI) {a(n)= if(n<1, n==0, -3*sumdiv(n, d, d*((d%6==1)+ (d%6==5)- 2*(d%6==3))))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)^3* eta(x^2+A)^3/ eta(x^3+A)/ eta(x^6+A), n))}
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CROSSREFS
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-3*A131944(n) = a(n) unless n = 0.
Sequence in context: A092103 A063550 A094152 this_sequence A100371 A100347 A165405
Adjacent sequences: A131940 A131941 A131942 this_sequence A131944 A131945 A131946
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Jul 30 2007
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