A255648 Expansion of (a(q) + a(q^2) + a(q^3) + a(q^6) - 4) / 6 in powers of q where a() is a cubic AGM theta function.

1, 1, 2, 1, 0, 2, 2, 1, 2, 0, 0, 2, 2, 2, 0, 1, 0, 2, 2, 0, 4, 0, 0, 2, 1, 2, 2, 2, 0, 0, 2, 1, 0, 0, 0, 2, 2, 2, 4, 0, 0, 4, 2, 0, 0, 0, 0, 2, 3, 1, 0, 2, 0, 2, 0, 2, 4, 0, 0, 0, 2, 2, 4, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 4, 2, 0, 2, 0, 0, 4, 0, 2, 0

Offset: 1

Author

Michael Somos, May 06 2015

Keywords

• nonn
• easy
• mult

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = q + q^2 + 2*q^3 + q^4 + 2*q^6 + 2*q^7 + q^8 + 2*q^9 + 2*q^12 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A033687, A035178, A248897.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

Programs

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ { 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0}[[Mod[ d, 18, 1]]], { d, Divisors[ n]}]];

PARI

{a(n) = if( n<1, 0, sumdiv(n, d, [ 0, 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1][d%18 + 1]))};

PARI

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 2, p%6==1, e+1, 1-e%2)))};

Formula

Expansion of (b(q^2)^2 / b(q) + b(q^6)^2 / b(q^3) - 2) / 3 in powers of q where b() is a cubic AGM theta function.

Expansion of (psi(q)^3 / psi(q^3) + psi(q^3)^3 / psi(q^9) - 2) / 3 in powers of q where psi() is a Ramanujan theta function.

Moebius transform is period 18 sequence [ 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0, ...].

a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2 if e>1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).

G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(2*k))^2 + (x^(3*k) + x^(9*k)) / (1 + x^(6*k))^2.

a(2*n) = a(n). a(3*n) = 2 * A035178(n). a(3*n + 1) = A033687(n). a(6*n + 5) = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Dec 22 2023