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A030204
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Expansion of q^-1*eta(q^8)*eta(q^16) in powers of q^8.
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5
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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
Euler transform of period 2 sequence [ -1,-2,...].
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REFERENCES
| M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
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FORMULA
| G.f.: Product_{k>0} (1-x^k)(1-x^(2k)).
a(9n+1)=-a(n), a(9n+4)=a(9n+7)=0. - Michael Somos Mar 17 2004
a(n)=b(8n+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)^(e/2) if p == 3 (mod 8) and e even, b(p^e)=1 if p == 5,7 (mod 8) and e even, b(p^e) = e+1 if p == 1 (mod 8) and p=x^2+32y^2, b(p^e) = (-1)^e*(e+1) if p == 1 (mod 8) and p is not of the form x^2+32y^2.
G.f.: (Sum_{k>0} x^((k^2-k)/2)) (Sum_{k} (-1)^k x^k^2) . - Michael Somos Sep 02 2006
Expansion of psi(q) * phi(-q) = f(-q^2) * f(-q) = f(-q)^2 / chi(-q) = f(-q)^3 / phi(-q) = f(-q^2)^2 * chi(-q) = f(-q^2)^3 / psi(q) = psi(-q) * phi(-q^2) = psi(q)^2 * chi(-q)^3 = phi(-q)^2 / chi(-q)^3 = (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. - Michael Somos, Mar 22 2008
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EXAMPLE
| eta(q^8)eta(q^16)=q -q^9 -2q^17 +q^25 +2q^41 +...
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^2+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(-4*p)%8, (-1)^e, 1), if(e%2==0, (-1)^(e/2*(p%8<5))))))))} /* Michael Somos Jul 26 2006 */
(PARI) {a(n)=if(n<0, 0, n=8*n+1; (qfrep([1, 0; 0, 32], n)-qfrep([4, 2; 2, 9], n))[n])} /* Michael Somos Sep 02 2006 */
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CROSSREFS
| Sequence in context: A190893 * A083650 A138514 A143540 A030200 A095734
Adjacent sequences: A030201 A030202 A030203 * A030205 A030206 A030207
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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