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Search: id:A093085
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| A093085 |
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Expansion of q^(1/2)eta(q)^2*eta(q^4)/(eta(q^2)*eta(q^8)^2) in powers of q. |
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| 1, -2, 0, 0, 1, 2, 0, 0, -1, -4, 0, 0, 0, 6, 0, 0, 1, -8, 0, 0, 0, 12, 0, 0, -1, -18, 0, 0, -1, 24, 0, 0, 2, -32, 0, 0, 1, 44, 0, 0, -2, -58, 0, 0, -1, 76, 0, 0, 2, -100, 0, 0, 1, 128, 0, 0, -3, -164, 0, 0, -1, 210, 0, 0, 4, -264, 0, 0, 2, 332, 0, 0, -5, -416, 0, 0, -2, 516, 0, 0, 5, -640, 0, 0, 2, 790, 0, 0, -6, -968
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OFFSET
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0,2
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COMMENT
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eta(q^2)eta(q^8)^6 = eta(q^1)^2eta(q^4)^2eta(q^8)eta(q^16)^2 +2eta(q^2)eta(q^4)^2eta(q^16)^4 is equivalent to the a(4n),...,a(4n+3) results.
G.f. A(x) satisfies 0=f(x/A(x)^2,x^2/A(x^2)^2) where f(u,v)=u^2-v-8uv-4v^2-32uv^2.
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FORMULA
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G.f.: Product_{k>0} (1-x^n)^2/((1-x^(4n-2))(1-x^(8n))^2).
Expansion of phi(-q) / psi(q^4) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 8 sequence [ -2, -1, -2, -2, -2, -1, -2, 0, ...].
Given g.f. A(x), then B(x) = A(x)^2 / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u * (u + 8) * (v + 4) - v^2.
G.f. is a Fourier series which satisfies f(-1/ (32 t)) = 32^(1/2) / f(t) where q = exp(2 pi i t).
a(4*n+2) = a(4*n+3) = 0.
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EXAMPLE
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1/q - 2*q + q^7 + 2*q^9 - q^15 - 4*q^17 + 6*q^25 + q^31 - 8*q^33 + ...
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PROGRAM
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(PARI) a(n)=if(n<0, 0, polcoeff(prod(k=1, n, (1-x^k)^[0, 2, 1, 2, 2, 2, 1, 2][1+k%8], 1+x*O(x^n)), n))
(PARI) a(n)=local(A, A2, m); if(n<0, 0, A=x+O(x^2); m=1; while(m<=n, m*=2; A=subst(A, x, x^2); A=4*A+16*A^2+(1+8*A)*sqrt(A+4*A^2)); polcoeff(sqrt(x/A), n))
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) / (eta(x^2 + A) * eta(x^8 + A)^2), n))}
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CROSSREFS
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Cf. A029838, A029839, A083365.
A029838(n) = a(4*n). -2 * A083365(n) = a(4*n+1). Convolution square is A131124.
Sequence in context: A025873 A112171 A112172 this_sequence A023555 A143377 A143380
Adjacent sequences: A093082 A093083 A093084 this_sequence A093086 A093087 A093088
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Mar 20 2004, Oct 22 2007
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