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A143378
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Expansion of q^(-1/24) * eta(q^2)^5 / eta(q) / eta(q^4)^2 in powers of q.
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3
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1, 1, -3, -2, 2, -1, 0, 1, 2, 4, -2, 0, -1, -2, 2, -1, 0, -2, -2, -2, 0, 0, 1, 4, -2, 2, 1, 0, -2, 0, 4, 0, 2, 0, 0, 1, 0, -4, 0, -2, -3, 0, 2, 2, -4, 0, 0, 2, -2, 0, -2, -3, 2, 0, 2, 2, 0, 1, 4, 0, 0, 0, 2, 0, 0, -4, 0, 2, 0, 2, -1, 0, 0, 2, -2, 2, -2, -1, -2, -4, 0, 0, 0, -2, -2, 0, 0, 2, 2, -2, 2, 0, 1, 0, 0, -2, 0, 0, 0, -2, 5, 2, -4, 2, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 phi(-q^2)^2 / chi(-q) in powers of q where phi(), chi() are Ramanujan theta functions.
Euler transform of period 4 sequence [ 1, -4, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 288^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A143380.
G.f.: Product_{k>0} (1 - (-x)^k)^2 * (1 - x^(2*k-1)).
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EXAMPLE
| q + q^25 - 3*q^49 - 2*q^73 + 2*q^97 - q^121 + q^169 + 2*q^193 + 4*q^217 + ...
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PROG
| (PARI) {a(n)= local(A, p, e, x); if(n<0, 0, n *= 4; 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^2 + A)^5 / eta(x + A) / eta(x^4 + A)^2, n))}
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CROSSREFS
| A143377(4*n) = A143380(4*n) = a(n).
Sequence in context: A144948 A108335 A155917 * A131961 A010269 A077450
Adjacent sequences: A143375 A143376 A143377 * A143379 A143380 A143381
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Aug 11 2008
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