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Search: id:A030207
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| A030207 |
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Expansion of eta(q)^2 * eta(q^2) * eta(q^4) * eta(q^8)^2 in powers of q. |
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5
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| 1, -2, -2, 4, 0, 4, 0, -8, -5, 0, 14, -8, 0, 0, 0, 16, 2, 10, -34, 0, 0, -28, 0, 16, 25, 0, 28, 0, 0, 0, 0, -32, -28, -4, 0, -20, 0, 68, 0, 0, -46, 0, 14, 56, 0, 0, 0, -32, 49, -50, -4, 0, 0, -56, 0, 0, 68, 0, -82, 0, 0, 0, 0, 64, 0, 56, 62, 8, 0, 0, 0, 40, -142, 0, -50, -136, 0, 0, 0, 0, -11, 92, 158, 0, 0, -28, 0
(list; graph; listen)
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OFFSET
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1,2
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REFERENCES
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M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
M. Koike, Matheiu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
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LINKS
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W. Stein, Modular Forms Database.
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FORMULA
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Expansion of q * phi(q) * phi(-q)^2 * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos May 28 2007
Euler transform of period 8 sequence [ -2, -3, -2, -4, -2, -3, -2, -6, ...]. - Michael Somos May 28 2007
Expansion of a newform level 8 weight 3 and character [1,1].
a(8n+5) = a(8n+7) = 0. a(2n) = -2*a(n).
G.f.: x * Product_{k>0} (1-x^k)^6 * (1+x^k)^4 * (1+x^(2k))^3 * (1+x^(4k))^6 . - Michael Somos May 28 2007
Expansion of (3* phi(q)^3* phi(q^2)^3 -2* phi(q)* phi(q^2)^5 -phi(q)^5* phi(q^2))/2 in powers of q where phi() is a Ramanujan theta function. - Michael Somos Jun 13 2007
a(n) is multiplicative with a(2^e) = (-2)^e, a(p^e) = (1+(-1)^e)/2 * p^e if p == 5, 7 (mod 8), a(p^e) = a(p)*a(p^(e-1)) - p^2*a(p^(e-2)) if p == 1, 3 (mod 8) where a(p) = 4*x^2 -2*p and p = x^2 +2*y^2. - Michael Somos Jun 13 2007
G.f.: (1/2)* Sum_{u,v} (u*u -2*v*v)* x^(u*u +2*v*v). - Michael Somos Jun 14 2007
G.f. is Fourier series of a weight 3 level 8 cusp form. f(-1/(8 t)) = i 2^(9/2) t^3 f(t) where q = exp(2 pi i t). - Michael Somos Jul 25 2007
Associated with permutations in Mathieu group M24 of shape (8)^2(4)(2)(1)^2.
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EXAMPLE
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q - 2*q^2 - 2*q^3 + 4*q^4 + 4*q^6 - 8*q^8 - 5*q^9 + 14*q^11 - 8*q^12 + ...
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PROGRAM
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(PARI) {a(n)= local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^8+A))^2*eta(x^2+A)*eta(x^4+A), n))} /* Michael Somos May 28 2007 */
(PARI) {a(n)= local(A, p, e, x, y, a0, a1); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, (-2)^e, if(p%8>4, if(e%2, 0, p^e), for(x=1, sqrtint(p\2), if(issquare(p-2*x^2, &y), break)); y=4*y^2-2*p; a0=1; a1=y; for(i=2, e, x=y*a1 -p^2*a0; a0=a1; a1=x); a1)))))} /* Michael Somos Jun 13 2007 */
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CROSSREFS
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Cf. A128712(n) = a(8n+1). -2*A128713(n) = a(8n+3).
Adjacent sequences: A030204 A030205 A030206 this_sequence A030208 A030209 A030210
Sequence in context: A049802 A129240 A127786 this_sequence A061006 A080736 A113750
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KEYWORD
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sign,mult
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AUTHOR
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njas
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