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A133739
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Expansion of eta(q^2)^23 * eta(q^3)^3 * eta(q^12)^6 / ( eta(q)^9 * eta(q^4)^10 * eta(q^6)^9 ) in powers of q.
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1
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1, 9, 31, 45, 6, -45, 8, 117, 121, 54, 12, -9, 14, 72, 186, 261, 18, -207, 20, 270, 248, 108, 24, 63, 31, 126, 391, 360, 30, -270, 32, 549, 372, 162, 48, -171, 38, 180, 434, 702, 42, -360, 44, 540, 726, 216, 48, 207, 57, 279, 558, 630, 54, -693, 72, 936, 620, 270, 60, -54
(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^6) / psi(q^3))^3 * phi(q)^5 / psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 9, -14, 6, -4, 9, -8, 9, -4, 6, -14, 9, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 18 (t/i)^2 g(t) where q = exp(2 pi i t) and g() is g.f. for A134078.
G.f.: f(x) + 6 * f(x^2) + 27 * f(x^3) + 20 * f(x^4) - 162 * f(x^6) + 108 * f(x^12) where f() is g.f. of A000203.
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EXAMPLE
| q + 9*q^2 + 31*q^3 + 45*q^4 + 6*q^5 - 45*q^6 + 8*q^7 + 117*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^2 + A)^23 * eta(x^3 + A)^3 * eta(x^12 + A)^6 / ( eta(x + A)^9 * eta(x^4 + A)^10 * eta(x^6 + A)^9 ), n))}
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CROSSREFS
| 9 * A134077(n) = a(4*n+2). 6 * A098098(n) = a(6*n+5).
Sequence in context: A161684 A054310 A072887 * A168297 A004126 A177342
Adjacent sequences: A133736 A133737 A133738 * A133740 A133741 A133742
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Oct 06 2007
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