A138507 Expansion of (f(q)^5 / f(q^5) - 1) / 5 in powers of q where f() is a Ramanujan theta function.

1, 1, -2, -3, 1, -2, -6, 5, 7, 1, 12, 6, -12, -6, -2, -11, -16, 7, 20, -3, 12, 12, -22, -10, 1, -12, -20, 18, 30, -2, 32, 21, -24, -16, -6, -21, -36, 20, 24, 5, 42, 12, -42, -36, 7, -22, -46, 22, 43, 1, 32, 36, -52, -20, 12, -30, -40, 30, 60, 6, 62, 32, -42, -43, -12, -24, -66, 48, 44, -6, 72, 35, -72, -36, -2, -60, -72, 24

Offset

1,3

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Links

Table of n, a(n) for n=1..78.

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

-(-1)^n * A109091(n) = a(n). A138506(n) = 5 * a(n) unless n=0.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A128255 A154948 A109091 * A209579 A205699 A109200

Adjacent sequences: A138504 A138505 A138506 * A138508 A138509 A138510

Keyword

sign,mult

Author

Michael Somos, Mar 21 2008

Status

approved