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A112301
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Expansion of (eta(q) * eta(q^16))^2 / (eta(q^2) * eta(q^8)) in powers of q.
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3
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1, -2, 0, 0, 2, 0, 0, 0, 1, -4, 0, 0, 2, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 3, -4, 0, 0, 2, 0, 0, 0, 0, -4, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, -6, 0, 0, 2, 0, 0, 0, 0, -4, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0, 1, -4, 0, 0, 4, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 2, 0, 0, 0, 0
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OFFSET
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1,2
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COMMENTS
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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
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Table of n, a(n) for n=1..105.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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Euler transform of period 16 sequence [ -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -2, ...].
Multiplicative with a(2)=-2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).
G.f.: Product_{k>0} (1-x^k)^2(1+x^(8k))^2(1+x^(2k))(1+x^(4k)).
Expansion of q * phi(-q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (phi(-q^2)^2 - phi(-q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.
Moebius transform is period 16 sequence [ 1, -3, -1, 2, 1, 3, -1, 0, 1, -3, -1, -2, 1, 3, -1, 0, ...].
G.f. is a period 1 Fourier series that satisfies f(-1 / (16 t)) = 4 (t/i) f(t) where q = exp(2 pi i t).
a(4n) = a(4n+3) = a(8n+6) = 0. a(8n+2) = -2 * a(4n+1).
G.f.: Sum_{k>0} kronecker(-4, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)).
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EXAMPLE
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q - 2*q^2 + 2*q^5 + q^9 - 4*q^10 + 2*q^13 + 2*q^17 - 2*q^18 + 3*q^25 - ...
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PROG
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(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^16+A))^2/(eta(x^2+A)*eta(x^8+A)), n))}
(PARI) {a(n) = if( n>0 & (n+1)%4\2, (n%2*3 - 2) * sumdiv( n/gcd(n, 2), d, (-1)^(d\2)))}
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CROSSREFS
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-(-1)^n * A134013(n) = a(n). A008441(n) = a(4*n+1). A113407(n) = a(8*n+1). 2 * A053692(n) = a(8*n+5).
Sequence in context: A185338 A208603 * A134013 A136521 A066448 A108497
Adjacent sequences: A112298 A112299 A112300 * A112302 A112303 A112304
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Sep 02 2005, Oct 02 2007
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STATUS
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approved
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