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Search: id:A129447
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| A129447 |
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Expansion of psi(q)* psi(q^3)* phi(q^3)/ phi(q) in powers of q where psi(), phi() are Ramanujan theta functions. |
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+0
2
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| 1, -1, 2, 0, 1, 0, 2, -2, 2, 0, 0, 0, 3, -1, 2, 0, 0, 0, 2, -2, 2, 0, 2, 0, 1, -2, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, -3, 0, 0, 1, 0, 4, -2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, -2, 2, 0, 2, 0, 1, -2, 4, 0, 0, 0, 0, -2, 2, 0, 0, 0, 4, -1, 2, 0, 2, 0, 2, -2, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, -2, 4, 0, 0, 0, 2, -4, 2, 0, 0, 0, 4, 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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Expansion of q^(-1/2)* eta(q)* eta(q^4)^2* eta(q^6)^7/ (eta(q^2)^3* eta(q^3)^3* eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ -1, 2, 2, 0, -1, -2, -1, 0, 2, 2, -1, -2, ...].
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 (mod 4), b(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).
a(6n+3)=a(6n+5)=0.
G.f.: Product_{k>0} (1+x^(2k))^2* (1-x^(3k))^2* (1+x^(3k))^5/ ((1+x^k)* (1+x^(6k))^2) .
G.f.: Sum_k x^(3k)/ (1 +x^(6k+1)) = Sum_{k>0} x^(k-1)* (1-x^(2k-1))^2/ (1+x^(6k-3)) .
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EXAMPLE
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q - q^3 + 2*q^5 + q^9 + 2*q^13 - 2*q^15 + 2*q^17 + 3*q^25 - q^27 +...
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PROGRAM
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(PARI) {a(n)=if(n<0, 0, if(n%6==1, n\=3; -1, 1)* sumdiv(2*n+1, d, kronecker(-4, d)) )}
(PARI) {a(n)=local(A, p, e); if(n<0, 0, n=2*n+1; A=factor(n); prod( k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p==3, (-1)^e, if(p%4==1, e+1, !(e%2)))))))}
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CROSSREFS
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A002175(n)= a(6n). A008441(n)= a(2n)= -a(6n+1). A121444(n)= a(6n+2)/2. A125079(n)= |a(n)|.
Sequence in context: A139353 A029397 A125079 this_sequence A104597 A072662 A030010
Adjacent sequences: A129444 A129445 A129446 this_sequence A129448 A129449 A129450
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Apr 16 2007
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