|
|
A111949
|
|
Expansion of eta(q) * eta(q^2) * eta(q^10) * eta(q^20) / (eta(q^4) * eta(q^5)) in powers of q.
|
|
3
|
|
|
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;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Number 37 of the 74 eta-quotients listed in Table I of Martin (1996).
|
|
LINKS
|
|
|
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)).
|
|
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
|
|
|
STATUS
|
approved
|
|
|
|