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Search: id:A161969
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| A161969 |
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Expansion of f(q)^8 in powers of q where f() is a Ramanujan theta function. |
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1
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| 1, 8, 20, 0, -70, -64, 56, 0, -125, 160, 308, 0, 110, 0, -520, 0, 57, -560, 0, 0, 182, 512, -880, 0, 1190, 448, 884, 0, 0, 0, -1400, 0, -1330, -1000, 1820, 0, -646, -1280, 0, 0, -1331, 2464, 380, 0, 1120, 0, 2576, 0, 0, 880, 1748, 0, -3850, 0, -3400, 0, 2703, -4160, -2500, 0, 3458
(list; graph; listen)
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OFFSET
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0,2
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FORMULA
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Expansion of q^(-1/3) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^8 in powers of q.
Euler transform of period 4 sequence [ 8, -16, 8, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1296 (t/i)^4 f(t) where q = exp(2 pi i t).
a(4*n + 3) = a(16*n + 13) = 0. a(4*n + 1) = (-1)^n * 8 * a(n).
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(2^e) = (1+(-1)^e)/2 * -(-8)^(e/2) if e>0, b(p^e) = (1+(-1)^e)/2 * (-p^3)^(e/2) if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^3 if p == 1 (mod 6) where b(p) = (x^2-3*p) * x, 4*p = x^2 + 3 * y^2, |x| < |y| and x == 2 (mod 3).
G.f.: Product_{k>0} (1 - (-x)^k)^8.
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EXAMPLE
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q + 8*q^4 + 20*q^7 - 70*q^13 - 64*q^16 + 56*q^19 - 125*q^25 + 160*q^28 + ...
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PROGRAM
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(PARI) {a(n) = if( n<0, 0, polcoeff( eta(-x + x*O(x^n))^8, n))};
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CROSSREFS
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A000731(n) = (-1)^n * a(n).
Sequence in context: A029845 A124972 A000731 this_sequence A034433 A120081 A081963
Adjacent sequences: A161966 A161967 A161968 this_sequence A161970 A161971 A161972
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Jun 22 2009
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EXTENSIONS
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Corrected by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Sep 02 2009
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