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Search: id:A143379
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| A143379 |
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Expansion of q^(-7/24) * eta(q) * eta(q^4)^2 / eta(q^2) in powers of q. |
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4
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| 1, -1, 0, -1, -1, 1, 1, 1, -1, 1, 0, 1, 0, 0, -2, -1, 0, 0, -1, 1, 1, -2, 0, 0, 0, 1, 1, 0, 2, 0, 1, -1, -1, 0, 1, -1, 0, 0, 1, 0, -1, -1, 0, -1, -1, -1, 0, 0, 0, 1, 0, 1, 0, 1, -1, -1, 2, 0, -1, 1, -1, 1, 0, 3, 1, -1, 0, 0, 0, 1, -2, 0, 0, -1, -1, 0, -1, 0, 1, 0, 0, 1, -1, -1, -1, 0, 0, 0, 0, -1, 0, -2, 0, 1, 2, 1, -1, 0, 2, 1, 0, 0, 0, 0, 1
(list; graph; listen)
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OFFSET
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0,15
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FORMULA
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Expansion of psi(-q)^2 / chi(-q) in powers of q where phi(), chi() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -1, 0, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 72^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A143377.
G.f.: Product_{k>0} (1 - x^(4*k))^2 * (1 - x^(2*k-1)).
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EXAMPLE
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q^7 - q^31 - q^79 - q^103 + q^127 + q^151 + q^175 - q^199 + q^223 + ...
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PROGRAM
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(PARI) {a(n)= local(A, p, e, x); if(n<0, 0, n = n*4 + 1; A = factor(6*n + 1); simplify( I^n / -2 * prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if(p<5, 0, if(p%8==5 | p%24==23, !(e%2), if(p%8==3 | p%24==17, (-1)^(e\2)*!(e%2), for(i=1, sqrtint(p\6), if( issquare(p - 6*i^2, &x), break)); (e+1) * (kronecker(12, x) * I^((p-1) / 6))^e)))))))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A), n))}
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CROSSREFS
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-A143377(4*n + 1) = A143380(4*n + 1) = 2 * a(n).
Sequence in context: A060953 A082858 A115953 this_sequence A136567 A109708 A035468
Adjacent sequences: A143376 A143377 A143378 this_sequence A143380 A143381 A143382
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Aug 11 2008
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