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A134747
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Expansion of q * (chi(-q) / chi(-q^4))^8 in powers of q where chi() is a Ramanujan theta function.
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0
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1, -8, 28, -64, 142, -352, 792, -1536, 2917, -5744, 10868, -19200, 33414, -58816, 101256, -167936, 275314, -452392, 732748, -1160064, 1819808, -2851104, 4421064, -6752256, 10236407, -15476272, 23215192, -34450944, 50811638, -74701632, 109138272
(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 k * (1 - k) / ( 4 * (1 + k) ) in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.
Expansion of ( (eta(q) * eta(q^8)) / (eta(q^2) * eta(q^4)) )^8 in powers of q.
Euler transform of period 8 sequence [ -8, 0, -8, 8, -8, 0, -8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = 16 * u*w * (v*w-1) * (v*u-1) - (v - u^2) * (v - w^2).
G.f. is Fourier series of a weight 0 level 8 modular form. f(-1/(8 t)) = f(t) where q = exp(2 pi i t).
G.f.: x * ( Product_{k>0} (1 + x^(4*k)) / (1 + x^k) )^8.
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EXAMPLE
| q - 8*q^2 + 28*q^3 - 64*q^4 + 142*q^5 - 352*q^6 + 792*q^7 - 1536*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 + A) * eta(x^8 + A)) / (eta(x^2 + A) * eta(x^4 + A)) )^8, n))}
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CROSSREFS
| Convolution invserse of A131123.
Sequence in context: A002408 A101127 A007259 * A083013 A028553 A100182
Adjacent sequences: A134744 A134745 A134746 * A134748 A134749 A134750
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Nov 07 2007
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