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A143377
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Expansion of q^(-1/6) * eta(q)^2 * eta(q^4) / eta(q^2) in powers of q.
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3
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1, -2, 0, 0, 1, 2, 0, 0, -3, 0, 0, 0, -2, 2, 0, 0, 2, 2, 0, 0, -1, -2, 0, 0, 0, -2, 0, 0, 1, -2, 0, 0, 2, 2, 0, 0, 4, -2, 0, 0, -2, 0, 0, 0, 0, -2, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 2, 4, 0, 0, -1, 2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -2, 2, 0, 0, -2, -2, 0, 0, 0, -2, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, -2, 0, 0, 0, 2, -2, 0, 0, 1
(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 psi(-q)^2 * chi(-q^2) in powers of q where psi(), chi() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -2, -1, -2, -2, ...].
a(n) = (-1)^(n / 2) * b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 8) or p == 23 (mod 24), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 3 (mod 8) or p == 17 (mod 24) and p>3, b(p^e) = (e+1) * s^e if p == 1, 7 (mod 24) where p = x^2 + 6*y^2 and s = kronecker(12, x) * (-1)^((p-1) / 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 4608^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A143379.
a(4*n + 2) = a(4*n + 3) = 0.
G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)).
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EXAMPLE
| q - 2*q^7 + q^25 + 2*q^31 - 3*q^49 - 2*q^73 + 2*q^79 + 2*q^97 + 2*q^103 + ...
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PROG
| (PARI) {a(n)= local(A, p, e, x); if(n<0, 0, A = factor(6*n + 1); simplify( I^n * 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)^2 * eta(x^4 + A) / eta(x^2 + A), n))}
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CROSSREFS
| (-1)^n * A143380(n) = a(n). A143378(n) = a(4*n). -2 * A143379(n) = a(4*n + 1).
Sequence in context: A093085 A023555 * A143380 A034950 A099584 A100563
Adjacent sequences: A143374 A143375 A143376 * A143378 A143379 A143380
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Aug 10 2008
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