OFFSET
1,3
COMMENTS
LINKS
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
a(n) is multiplicative with a(2^e) = ((-2)^(e+1) - 1) / 3, a(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 7 (mod 10), a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10).
G.f.: (Product_{k>0} (1 - (-x)^k)^5 / (1 - (-x)^(5*k)) - 1) / 5.
L.g.f.: log(1/(1 - x/(1 + x^2/(1 - x^3/(1 + x^4/(1 - x^5/(1 + ...))))))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(15*sqrt(5)) = 0.294254... . - Amiram Eldar, Jan 29 2024
EXAMPLE
q + q^2 - 2*q^3 - 3*q^4 + q^5 - 2*q^6 - 6*q^7 + 5*q^8 + 7*q^9 + ...
PROG
(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, d * kronecker(5, d)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(-x + A)^5 / eta(-x^5 + A) - 1) / 5, n))}
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Michael Somos, Mar 21 2008
STATUS
approved