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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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6
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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
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OFFSET
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1,2
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COMMENTS
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Unique cusp form of weight 3 for congruence group Gamma_1(8). - Michael Somos, Aug 11 2011
Associated with permutations in Mathieu group M24 of shape (8)^2(4)(2)(1)^2.
For n nonzero, a(n) is nonzero if and only if n is in A002479.
Number 20 of the 74 eta-quotients listed in Table I of Martin (1996).
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LINKS
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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
Expansion of (3 * phi(q)^3 * phi(q^2)^3 - 2 * phi(q) * phi(q^2)^5 - phi(q)^5 * phi(q^2)) / 4 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jun 13 2007
Euler transform of period 8 sequence [-2, -3, -2, -4, -2, -3, -2, -6, ...]. - Michael Somos, May 28 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. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 512^(1/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 25 2007
G.f.: (1/2) * Sum_{u,v in Z} (u*u - 2*v*v) * x^(u*u + 2*v*v). - Michael Somos, Jun 14 2007
G.f.: x * Product_{k>0} (1 - x^k)^6 * (1 + x^k)^4 * (1 + x^(2*k))^3 * (1 + x^(4*k))^6. - Michael Somos, May 28 2007
a(8*n + 5) = a(8*n + 7) = 0. a(2*n) = -2*a(n). a(8*n + 1) = A128712(n). a(8*n + 3) = -2 * A128713(n).
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EXAMPLE
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G.f. = 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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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4] QPochhammer[ q^8]^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 3, 0, q^2] EllipticTheta[ 2, 0, q^2]^2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (-EllipticTheta[ 3, 0, q]^5 EllipticTheta[ 3, 0, q^2] + 3 EllipticTheta[ 3, 0, q]^3 EllipticTheta[ 3, 0, q^2]^3 - 2 EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2]^5) / 4, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
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PROG
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(PARI) {a(n) = my(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) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-2)^e, 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 */
(Sage) CuspForms( Gamma1(8), 3, prec = 100).0; # Michael Somos, Aug 11 2011
(Magma) Basis( CuspForms( Gamma1(8), 3), 100) [1]; /* Michael Somos, May 27 2014 */
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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