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A138505
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Expansion of ((phi(q) * phi(-q^2)^2)^2 - 1) / 4 in powers of q where phi() is a Ramanujan theta function.
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1
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1, -1, -8, -1, 26, 8, -48, -1, 73, -26, -120, 8, 170, 48, -208, -1, 290, -73, -360, -26, 384, 120, -528, 8, 651, -170, -656, 48, 842, 208, -960, -1, 960, -290, -1248, -73, 1370, 360, -1360, -26, 1682, -384, -1848, 120, 1898, 528, -2208, 8, 2353, -651, -2320, -170, 2810, 656, -3120, 48, 2880
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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
| a(n) is multiplicative with a(2^e) = -1 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) / ( -p^2 - 1) if p == 3 (mod 4).
a(2*n) = (-1)^n * a(n).
G.f.: Sum_{k>0} -(-1)^k * (2*k-1)^2 * x^(2*k-1) / (1 + x^(2*k-1)).
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EXAMPLE
| q - q^2 - 8*q^3 - q^4 + 26*q^5 + 8*q^6 - 48*q^7 - q^8 + 73*q^9 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-4, d) * -(-1)^(n/d)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^2 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^4))^2 - 1) / 4, n))}
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CROSSREFS
| Cf. A138504(n) = 4 * a(n).
Sequence in context: A183892 A019432 A002173 * A050458 A125166 A075155
Adjacent sequences: A138502 A138503 A138504 * A138506 A138507 A138508
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Mar 21 2008
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