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

0,2

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

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.

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

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

Cf. A033716, A093602, A096936, A112298, A122861.

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

Sequence in context: A246863 A227864 A236603 * A122861 A326045 A277097

Adjacent sequences: A129573 A129574 A129575 * A129577 A129578 A129579

Keyword

nonn,easy

Author

Michael Somos, Apr 23 2007

Status

approved