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A129391
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Expansion of phi(-q)* phi(q^5)/ (chi(-q^2)* chi(-q^10)) in powers of q.
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2
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1, -2, 1, -2, 3, 0, 0, -2, 0, 0, 4, -2, 1, -4, 2, 0, 0, -2, 0, 0, 2, -2, 3, -2, 3, 0, 0, 0, 0, 0, 2, -6, 0, -2, 4, 0, 0, -2, 0, 0, 5, -2, 0, -4, 2, 0, 0, 0, 0, 0, 2, -2, 4, -2, 2, 0, 0, -2, 0, 0, 1, -4, 1, -2, 4, 0, 0, -4, 0, 0, 4, 0, 2, -6, 2, 0, 0, 0, 0, 0, 4, -2, 0, -2, 1, 0, 0, -2, 0, 0, 2, -4, 0, 0, 8, 0, 0, 0, 0, 0, 4, -4, 2, -6, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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FORMULA
| Expansion of q^(-1/2)* eta(q)^2* eta(q^4)* eta(q^10)^4/ (eta(q^2)^2* eta(q^5)^2* eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ -2, 0, -2, -1, 0, 0, -2, -1, -2, -2, -2, -1, -2, 0, 0, -1, -2, 0, -2, -2, ...].
a(n)= b(2*n+1) where b(n) is multiplicative with b(2^e) = 0, b(5^e) = 1, b(p^e) = (-1)^e* (e+1) if p == 3, 7 (mod 20), b(p^e) = e+1 if p == 1, 9 (mod 20), b(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20).
G.f.: Sum_{k>0} a(k)* x^(2*k-1) = Sum_{k>0} (-1)^k* f(x^(2*k-1)) where f(x)= x* (1-x^2)* (1-x^6)/ (1-x^10).
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EXAMPLE
| q - 2*q^3 + q^5 - 2*q^7 + 3*q^9 - 2*q^15 + 4*q^21 - 2*q^23 + q^25 - ...
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PROG
| (PARI) {a(n)= if(n<0, 0, (-1)^n* sumdiv(2*n+1, d, kronecker(-20, 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==5, 1, if(p%20 <10, (-1)^(((p%20)%4==3)*e)* (e+1), !(e%2))))))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)^2* eta(x^4+A)* eta(x^10+A)^4/ (eta(x^2+A)^2* eta(x^5+A)^2* eta(x^20+A)), n))}
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CROSSREFS
| (-1)^n* A129390(n)= a(n).
Sequence in context: A127510 A158810 A129390 * A123590 A092872 A141455
Adjacent sequences: A129388 A129389 A129390 * A129392 A129393 A129394
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Apr 13 2007
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