OFFSET
1,3
COMMENTS
Number 37 of the 74 eta-quotients listed in Table I of Martin (1996).
LINKS
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
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, ...].
a(n) is 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^(2*k)) / (1 - x^(5*k)).
G.f.: Sum_{k>0} Kronecker(k, 5) * x^k / (1 + x^(2*k)).
G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(5*k)) * (1 - x^(20*k)) / (1 + x^(2*k)).
a(5*n) = a(n). - Michael Somos, May 19 2015
EXAMPLE
G.f. = q - q^2 - 2*q^3 + q^4 + q^5 + 2*q^6 - 2*q^7 - q^8 + 3*q^9 - q^10 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^10] QPochhammer[ q^20] / (QPochhammer[ q^4] QPochhammer[ q^5]), {q, 0, n}]; (* Michael Somos, May 19 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[d, 2] (-1)^Quotient[d, 2] KroneckerSymbol[ n/d, 5], { d, Divisors[ n]}]]; (* Michael Somos, May 19 2015 *)
PROG
(PARI) {a(n) = my(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])};
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Michael Somos, Aug 22 2005
STATUS
approved