A185717 Expansion of q^(-1) * c(q^2) * (c(q) - c(q^4)) / 9 in powers of q^2 where c() is a cubic AGM theta function.

1, 3, 6, 8, 9, 12, 14, 18, 18, 20, 24, 24, 31, 27, 30, 32, 36, 48, 38, 42, 42, 44, 54, 48, 57, 54, 54, 72, 60, 60, 62, 72, 84, 68, 72, 72, 74, 93, 96, 80, 81, 84, 108, 90, 90, 112, 96, 120, 98, 108, 102, 104, 144, 108, 110, 114, 114, 144, 126, 144, 133, 126, 156, 128

Offset: 0

Author

Michael Somos, Feb 10 2011

Keywords

• nonn
• easy

Comments

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

Examples

			1 + 3*x + 6*x^2 + 8*x^3 + 9*x^4 + 12*x^5 + 14*x^6 + 18*x^7 + 18*x^8 + ...
q + 3*q^3 + 6*q^5 + 8*q^7 + 9*q^9 + 12*q^11 + 14*q^13 + 18*q^15 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A078708, A100044, A118271, A121443, A124449.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

A185717[n_] := SeriesCoefficient[(QPochhammer[q^3, q^3]/QPochhammer[-q^3, q^3])^4*(1/(QPochhammer[q, q^2]*QPochhammer[q^3, q^6])^3), {q, 0, n}];
Table[A185717[n], {n, 0, 50}] (* G. C. Greubel, Jul 10 2017 *)

PARI

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, if( (n/d) % 3, 1, 0) * d))}

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^5 / (eta(x + A)^3 * eta(x^6 + A)), n))}

Formula

Expansion of phi(-x^3)^4 / (chi(-x) * chi(-x^3))^3 in powers of x where phi(), chi() are Ramanujan theta functions.

Euler transform of period 6 sequence [ 3, 0, -2, 0, 3, -4, ...].

a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 3^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A118271.

a(3*n + 1) = 3 * a(n). A078708(2*n + 1) = A121443(2*n + 1) = A124449(2*n + 1).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/9 = 1.0966227... (A100044). - Amiram Eldar, Dec 28 2023