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A111949
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Expansion of eta(q)eta(q^2)et(q^10)eta(q^20)/(eta(q^4)eta(q^5)) in powers of q.
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1
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1, -1, -2, 1, 1, 2, -2, -1, 3, -1, 0, -2, 0, 2, -2, 1, 0, -3, 0, 1, 4, 0, -2, 2, 1, 0, -4, -2, 2, 2, 0, -1, 0, 0, -2, 3, 0, 0, 0, -1, 2, -4, -2, 0, 3, 2, -2, -2, 3, -1, 0, 0, 0, 4, 0, 2, 0, -2, 0, -2, 2, 0, -6, 1, 0, 0, -2, 0, 4, 2, 0, -3, 0, 0, -2, 0, 0, 0, 0, 1, 5, -2, -2, 4, 0, 2, -4, 0, 2, -3, 0, -2, 0, 2, 0, 2, 0, -3, 0, 1, 2, 0, -2, 0, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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FORMULA
| Euler transform of period 20 sequence [ -1, -2, -1, -1, 0, -2, -1, -1, -1, -2, -1, -1, -1, -2, 0, -1, -1, -2, -1, -2, ...].
Multiplicative with a(p^e) = (-1)^e if p=2, a(p^e) = 1 if p=5, a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20), a(p^e) = e+1 if p == 1, 9 (mod 20), a(p^e) = (e+1)(-1)^e if p == 3, 7 (mod 20).
G.f.: Sum_{k>0} kronecker(-4, k) x^k(1-x^k)(1-x^(2k))/(1-x^(5k)).
G.f.: Sum_{k>0} kronecker(k, 5)*x^k/(1+x^(2k)) = x Product_{k>0} (1-x^k)(1+x^(5k))*(1-x^(20k))/(1+x^(2k)).
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PROG
| (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^2+A)*eta(x^10+A)*eta(x^20+A)/eta(x^4+A)/eta(x^5+A), n))}
(PARI) {a(n)=if(n<1, 0, sumdiv(n, d, (d%2)* (-1)^(d\2)* kronecker(n/d, 5)))}
(PARI) {a(n)=if(n<1, 0, qfrep([1, 0; 0, 5], n)[n] -qfrep([2, 1; 1, 3], n)[n])}
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CROSSREFS
| Cf. A035170(n)=|a(n)|.
Sequence in context: A124233 A035170 * A143323 A086598 A074746 A133188
Adjacent sequences: A111946 A111947 A111948 * A111950 A111951 A111952
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Aug 22 2005
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