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A138501
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Expansion of (eta(q)^2 * eta(q^4)^4 / eta(q^2)^3)^2 in powers of q.
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1
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1, -4, 8, -16, 26, -32, 48, -64, 73, -104, 120, -128, 170, -192, 208, -256, 290, -292, 360, -416, 384, -480, 528, -512, 651, -680, 656, -768, 842, -832, 960, -1024, 960, -1160, 1248, -1168, 1370, -1440, 1360, -1664, 1682, -1536, 1848, -1920, 1898, -2112, 2208, -2048, 2353, -2604
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of q * (phi(-q) * psi(q^2)^2)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -4, 2, -4, -6, ...].
a(n) is multiplicative with a(2^e) = -(4^e) if e>0, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 4), a(p^e) = ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1) if p == 3 (mod 4).
G.f. is a Fourier series which satisfies f(-1/(4 t)) = 2 (t/i)^3 g(t) where q = exp(2 pi i t) and g() is g.f. for A138502
G.f.: x * (Product_{k>0} (1 - x^k)^3 * (1 + x^k) * (1 + x^(2*k))^4)^2.
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EXAMPLE
| q - 4*q^2 + 8*q^3 - 16*q^4 + 26*q^5 - 32*q^6 + 48*q^7 - 64*q^8 + 73*q^9 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, d^2 * (n / d % 2) * (-1)^(n / d \ 2)))}
(PARI) {a(n) = local(A, p, e, f); if( n<0, 0, A = factor(n); prod(k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==2, -4^e, f = (-1)^(p\2); ((p^2)^(e+1) - f^(e+1)) / (p^2 - f)))))}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n) ; polcoeff( (eta(x + A)^2 * eta(x^4 + A)^4 / eta(x^2 + A)^3)^2, n))}
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CROSSREFS
| -(-1)^n * A050470(n) = a(n).
Sequence in context: A013934 A167189 A050470 * A096296 A068936 A054744
Adjacent sequences: A138498 A138499 A138500 * A138502 A138503 A138504
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Mar 20 2008
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