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A134015
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Expansion of (1 - phi(-q) * phi(q^4)) / 2 in powers of q where phi() is a Ramanujan theta function.
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1
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1, 0, 0, -2, 2, 0, 0, -2, 1, 0, 0, 0, 2, 0, 0, -2, 2, 0, 0, -4, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, -2, 2, 0, 0, -4, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, -4, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2, 4, 0, 0, -4, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, -4, 1, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -6, 2, 0, 0, -4, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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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
| Moebius transform is period 16 sequence [ 1, -1, -1, -2, 1, 1, -1, 0, 1, -1, -1, 2, 1, 1, -1, 0, ...].
Multiplicative with a(2) = 0, a(2^e) = -2 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).
a(4*n+2) = a(4*n+3) = 0.
G.f.: x/(1+x^2) + x^3/(1+x^6) - 2 * x^4/(1+x^8) + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, if( n%4 < 2, (n%2*3 - 2) * sumdiv(n, d, kronecker(-4, d))))}
(PARI) {a(n) = -(-1)^n * if( n<1, 0, qfrep([1, 0; 0, 4], n)[n])}
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CROSSREFS
| -(-1)^n * A113406(n) = a(n). A134014(n) = -2 * a(n) unless n=0. -2 * A002654(n) = a(4*n). A008441(n) = a(4*n+1).
Sequence in context: A062070 A179851 A113406 * A151851 A033461 A143432
Adjacent sequences: A134012 A134013 A134014 * A134016 A134017 A134018
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Oct 02 2007
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