A129576 Expansion of phi(x) * chi(x) * psi(-x^3) in powers of x where phi(), chi(), psi() are Ramanujan theta functions.

1, 3, 2, 0, 2, 3, 2, 0, 1, 6, 2, 0, 2, 0, 2, 0, 3, 6, 0, 0, 2, 3, 2, 0, 2, 6, 2, 0, 0, 0, 4, 0, 2, 3, 2, 0, 2, 6, 0, 0, 1, 6, 2, 0, 4, 0, 2, 0, 0, 6, 2, 0, 2, 0, 2, 0, 3, 6, 2, 0, 2, 0, 0, 0, 2, 9, 2, 0, 0, 6, 2, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, 6, 4, 0, 0, 3, 4

Offset: 0

Author

Michael Somos, Apr 23 2007

Keywords

• nonn
• easy

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). - Michael Somos, Jun 28 2017

Examples

			G.f. = 1 + 3*x + 2*x^2 + 2*x^4 + 3*x^5 + 2*x^6 + x^8 + 6*x^9 + 2*x^10 + ...
G.f. = q + 3*q^4 + 2*q^7 + 2*q^13 + 3*q^16 + 2*q^19 + q^25 + 6*q^28 + ...
		

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. A033716, A093602, A096936, A112298, A122861.

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

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := Length @ FindInstance[ x^2 + 3 y^2 == 3 n + 1, {x, y}, Integers, 10^9] / 2; (* Michael Somos, Sep 03 2016 *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# == 3, Boole[#2 == 0], # == 2, 3 (1 + (-1)^#2)/2, Mod[#, 3] == 2, (1 + (-1)^#2)/2, True, #2 + 1] & @@@ FactorInteger[3 n + 1])]; (* Michael Somos, Jun 28 2017 *)

PARI

{a(n) = if( n<0, 0, n = 3*n + 1; sumdiv(n, d, kronecker(-3, d) * [0, 1, -2, 1] [n/d%4 + 1] ))};

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)), n))};

PARI

{a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 3*(1-e%2), p==3, 0, p%3==2, 1-e%2, e+1)))}; /* Michael Somos, Jun 28 2017 */

Formula

From Michael Somos, Jun 28 2017: (Start)

Expansion of q^(-1/3) * (2*c(q) + c(-q)) / 3 = q^(-1/3) * (c(q) + 2*c(q^4)) / 3 in powers of q where c() is a cubic AGM theta function.

Expansion of (a(q) - a(q^3) + 2*a(q^4) - 2*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.

Expansion of q^(-1/3) * eta(q^2)^7 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)) in powers of q. (End)

Euler transform of period 12 sequence [3, -4, 2, -1, 3, -4, 3, -1, 2, -4, 3, -2, ...].

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

a(n) = A096936(3*n + 1) = A112298(3*n + 1).

2 * a(n) = A033716(3*n + 1). - Michael Somos, Sep 03 2016

a(n) = (-1)^n * A122161(n). - Michael Somos, Jun 28 2017

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Dec 29 2023