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Search: id:A128580
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| A128580 |
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Expansion of phi(q^3)* psi(q^4) -q* phi(q)* psi(q^12) in powers of q where phi(), psi() are Ramanujan theta functions. |
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+0
5
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| 1, -1, -2, 2, 1, -2, 0, 2, 0, 0, -2, 0, 3, -1, -2, 2, 2, -4, 0, 0, 0, 0, -2, 0, 3, 0, -2, 4, 0, -2, 0, 2, 0, 0, 0, 0, 2, -3, -4, 2, 1, -2, 0, 2, 0, 0, -2, 0, 2, -2, -2, 2, 4, -2, 0, 0, 0, 0, 0, 0, 3, 0, -4, 2, 0, -2, 0, 2, 0, 0, 0, 0, 4, -3, -2, 2, 0, -4, 0, 2, 0, 0, -4, 0, 1, 0, -2, 6, 2, -2, 0, 0, 0, 0, -2, 0, 2, 0, -2, 2, 0, -4, 0, 0, 0
(list; graph; listen)
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OFFSET
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0,3
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FORMULA
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a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1)* (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
Euler transform of period 24 sequence [ -1, -2, 0, 0, -1, -1, -1, -1, 0, -2, -1, -2, -1, -2, 0, -1, -1, -1, -1, 0, 0, -2, -1, -2, ...].
a(12n+6)= a(12n+8)= a(12n+9)= a(12n+11)= 0.
G.f.: Product_{k>0} (1-x^(8k))* (1-x^(12k))^2/ ((1+x^k)* (1+x^(2k))^2* (1-x^(3k))* (1+x^(12k))).
G.f.: Sum_{k>=0} a(k)*x^(2*k+1) = Sum_{k>0} (x^k - x^(3*k))/(1 + x^(4*k))* kronecker(-12, k) = Sum_{k>0} (x^k + x^(3*k))/(1 + x^(2*k) + x^(4*k))* kronecker(8, k).
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PROGRAM
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(PARI) {a(n)= if(n<0, 0, n=2*n+1; sumdiv(n, d, kronecker(-12, d)* kronecker(8, n/d)))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)* eta(x^2+A)* eta(x^8+A)* eta(x^12+A)^3/ (eta(x^3+A)* eta(x^4+A)^2* eta(x^24+A)), n))}
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CROSSREFS
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A128581(2n+1)= a(n). A128582(n)= -a(3n+2)/2. A113780(n)= a(12n).
Sequence in context: A145783 A094022 A145785 this_sequence A129402 A134177 A104405
Adjacent sequences: A128577 A128578 A128579 this_sequence A128581 A128582 A128583
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Mar 11 2007
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