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Search: id:A138502
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| A138502 |
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Expansion of q^(-1/2) * (eta(q)^4 * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q. |
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+0
2
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| 1, -8, 26, -48, 73, -120, 170, -208, 290, -360, 384, -528, 651, -656, 842, -960, 960, -1248, 1370, -1360, 1682, -1848, 1898, -2208, 2353, -2320, 2810, -3120, 2880, -3480, 3722, -3504, 4420, -4488, 4224, -5040, 5330, -5208, 5760, -6240, 5905, -6888, 7540, -6736, 7922, -8160, 7680
(list; graph; listen)
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OFFSET
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0,2
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FORMULA
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Expansion of (phi(-q)^2 * psi(q^2))^2 in powers of q where phi(), psi() are Ramanujan theta functons.
Euler transform of period 4 sequence [ -8, -2, -8, -6, ...].
a(n) = b(2*n+1) where b(n) is multiplicative where b(2^e) = 0^e, b(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 4), b(p^e) = (-(-p^2)^(e+1) + 1) / (p^2 + 1) if p == 3 (mod 4).
G.f. is a Fourier series which satisfies f(-1/(16 t)) = 256 (t/i)^3 g(t) where q = exp(2 pi i t) and g() is g.f. for A138501.
G.f.: (Product_{k>0} (1 - x^k)^3 * (1 + x^(2*k))^2 / (1 + x^k))^2.
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EXAMPLE
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q - 8*q^3 + 26*q^5 - 48*q^7 + 73*q^9 - 120*q^11 + 170*q^13 - 208*q^15 + ...
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PROGRAM
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(PARI) {a(n) = if(n<0, 0, n = 2*n+1; sumdiv(n, d, d^2 * kronecker(-4, d)))}
(PARI) {a(n) = local(A, p, e, f); if( n<0, 0, n = 2*n+1; A = factor(n); prod(k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==2, 0, f = (-1)^(p\2); f * ((f*p^2)^(e+1) - 1) / (p^2 - f)))))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n) ; polcoeff( (eta(x + A)^4 * eta(x^4 + A)^2 / eta(x^2 + A)^3)^2, n))}
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CROSSREFS
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(-1)^n * A122854(n) = A002173(2*n+1) = a(n).
Sequence in context: A031085 A031307 A122854 this_sequence A143894 A126176 A074238
Adjacent sequences: A138499 A138500 A138501 this_sequence A138503 A138504 A138505
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Mar 20 2008
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