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A128581
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Expansion of (phi(q^2)* phi(-q^3)- phi(-q)* phi(q^6))/2 in powers of q where phi() is a Ramanujan theta function.
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1
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1, 1, -1, -1, -2, -1, 2, 1, 1, -2, -2, 1, 0, 2, 2, -1, 0, 1, 0, 2, -2, -2, 0, -1, 3, 0, -1, -2, -2, 2, 2, 1, 2, 0, -4, -1, 0, 0, 0, -2, 0, -2, 0, 2, -2, 0, 0, 1, 3, 3, 0, 0, -2, -1, 4, 2, 0, -2, -2, -2, 0, 2, 2, -1, 0, 2, 0, 0, 0, -4, 0, 1, 2, 0, -3, 0, -4, 0, 2, 2, 1, 0, -2, 2, 0, 0, 2, -2, 0, -2, 0, 0, -2, 0, 0, -1, 2, 3, -2, -3, -2, 0, 2, 0, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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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 eta(q^2)^3* eta(q^3)* eta(q^12)* eta(q^24)/ (eta(q)* eta(q^6)^2* eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ 1, -2, 0, -2, 1, -1, 1, -1, 0, -2, 1, -2, 1, -2, 0, -1, 1, -1, 1, -2, 0, -2, 1, -2, ...].
Multiplicative with a(2^e) = -(-1)^e if e>0, a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1)(-1)^e if p == 5, 11 (mod 24), a(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
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EXAMPLE
| q + q^2 - q^3 - q^4 - 2*q^5 - q^6 + 2*q^7 + q^8 + q^9 - 2*q^10 - ...
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PROG
| (PARI) {a(n)= if(n<1, 0, -(-1)^n* sumdiv(n, d, kronecker(d, 8)* kronecker(n/d, 3)))}
(PARI) {a(n)= local(A); if(n<1, 0, n--; A= x*O(x^n); polcoeff( eta(x^2+A)^3* eta(x^3+A)* eta(x^12+A)* eta(x^24+A)/ (eta(x+A)* eta(x^6+A)^2* eta(x^8+A)), n))}
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CROSSREFS
| A115660(n)= -(-1)^n*a(n) = a(2n). A128580(n)= a(2n+1).
Sequence in context: A190611 A115660 * A192013 A026517 A072047 A106802
Adjacent sequences: A128578 A128579 A128580 * A128582 A128583 A128584
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Mar 11 2007
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