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A138514
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Expansion of q^(-1/8) * eta(q^2)^4 / (eta(q) * eta(q^4)) in powers of q.
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4
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1, 1, -2, -1, 0, -2, 1, 0, 0, 2, 1, 2, -2, 0, 2, 1, 0, -2, 0, -2, 0, -1, 0, 0, -2, 0, 0, 0, -1, 2, -2, 0, 2, 0, 0, 2, 3, 0, 0, -2, 0, 0, 2, 0, 2, 1, -2, 0, 0, 0, -2, -2, 0, 2, -2, 1, -2, -2, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, -2, 2, 0, 2, 2, 0, 2, 1, 0, -2, 0, 2, 0, -2, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, -2, -2, 0, 0, 0, 2, -2, 0, 0, -2
(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).
A030204, A083650 and A138514 are the same except for signs. - N. J. A. Sloane, May 07 2010
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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 f(q) * f(-q^2) = psi(-q) * phi(q) = chi(q) * f(-q^2)^2 = psi(q) * phi(-q^2) = f(q)^2 / chi(q) = f(q)^3 / phi(q) = f(-q^2)^3 / psi(-q) = phi(q)^2 / chi(q)^3 = chi(q)^3 * psi(-q)^2 = (f(q)^3 * psi(-q))^(1/2) = (f(-q^2)^3 * phi(q))^(1/2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ 1, -3, 1, -2, ...].
a(n) = b(8*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 0 if p == 3, 5, 7 (mod 8) and e odd, b(p^e) = 1 if p == 3 (mod 4) and e even, b(p^e) = (-1)^(e/2) if p == 5 (mod 8) and e even, b(p^e) = e+1 if p == 1 (mod 8) and p = x^2 + 64*y^2, b(p^e) = (-1)^e * (e+1) if p == 1 (mod 8) and p is not of the form x^2 + 64*y^2.
a(9*n+1) = a(n), a(9*n+4) = a(9*n+7) = 0.
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k-1)).
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EXAMPLE
| q + q^9 - 2*q^17 - q^25 - 2*q^41 + q^49 + 2*q^73 + q^81 + 2*q^89 - 2*q^97 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / eta(x + A) / eta(x^4 + A), n))}
(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if( p==2, 0, if( p%8==1, (e+1) * if( qfbclassno( -8 * p) / 4 % 2, (-1)^e, 1), if( e%2==0, (-1)^(e/2 * (p%8==5)))))))) }
(PARI) {a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([1, 0; 0, 64], n) - qfrep([4, 2; 2, 17], n))[n])}
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CROSSREFS
| (-1)^n * A030204(n) = a(n).
Sequence in context: A190893 A030204 A083650 * A143540 A030200 A095734
Adjacent sequences: A138511 A138512 A138513 * A138515 A138516 A138517
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Mar 22 2008
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