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A132974
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Expansion of psi(-q^3) / psi(-q)^3 in powers of q where psi() is a Ramanujan theta function.
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5
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1, 3, 6, 12, 24, 45, 78, 132, 222, 363, 576, 900, 1392, 2121, 3180, 4716, 6936, 10098, 14550, 20796, 29520, 41595, 58176, 80856, 111750, 153561, 209820, 285240, 385968, 519840, 696960, 930516, 1237470, 1639314, 2163456, 2845080, 3728904, 4871211
(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 eta(q^2)^3 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6) ) in powers of q.
Euler transform of period 12 sequence [ 3, 0, 2, 3, 3, 0, 3, 3, 2, 0, 3, 2, ...].
G.f.: Product_{k>0} (1-x^(3*k)) * (1+x^(6*k)) / ( (1-x^k) * (1+x^(2*k)) )^3.
G.f. is a period 1 Fourier series which satisfies f(-1/(12t)) = (108)^(-1/2) (t/i)^(-1) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A133637.
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EXAMPLE
| 1 + 3*q + 6*q^2 + 12*q^3 + 24*q^4 + 45*q^5 + 78*q^6 + 132*q^7 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A ) / ( eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A) ), n))}
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CROSSREFS
| A132979(n) = (-1)^n * a(n). Convolution inverse of A132973.
Sequence in context: A039695 A079079 * A132979 A163314 A018183 A196787
Adjacent sequences: A132971 A132972 A132973 * A132975 A132976 A132977
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Sep 07 2007
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