A214361 Expansion of c(q^2) * (c(q) + 2 * c(q^4)) / 9 in powers of q where c() is a cubic AGM theta function.

1, 3, 3, 3, 6, 9, 8, 3, 9, 18, 12, 9, 14, 24, 18, 3, 18, 27, 20, 18, 24, 36, 24, 9, 31, 42, 27, 24, 30, 54, 32, 3, 36, 54, 48, 27, 38, 60, 42, 18, 42, 72, 44, 36, 54, 72, 48, 9, 57, 93, 54, 42, 54, 81, 72, 24, 60, 90, 60, 54, 62, 96, 72, 3, 84, 108, 68, 54

Offset: 1

Author

Michael Somos, Jul 18 2012

Keywords

• nonn
• easy
• mult

Comments

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

b(n) = 6*a(n) is the number of solutions in integers to n = x^2 + y^2 + z^2 + w^2 where x + y + z is not divisible by 3. - Michael Somos, Jun 23 2018

Examples

			G.f. = q + 3*q^2 + 3*q^3 + 3*q^4 + 6*q^5 + 9*q^6 + 8*q^7 + 3*q^8 + 9*q^9 + 18*q^10 + ...
a(1) = 1, b(1) = 6 with solutions (w, x, y, z) = {(0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0)} and their negatives. - _Michael Somos_, Jun 23 2018
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Michael Somos, Introduction to Ramanujan theta functions.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A086463, A121443, A124449, A185717, A214456.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 - EllipticTheta[ 3, 0, q^3]^4) / 8, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] QPochhammer[ -q^3, q^6])^3 EllipticTheta[ 2, 0, (-q)^(3/2)]^4 / (-16 (-q)^(1/2)), {q, 0, n}];

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^3 + A) * eta(x^6 + A)^2 * eta(x^12 + A) / ( eta(x + A) * eta(x^4 + A))^3, n))};

PARI

{a(n) = if( n<1, 0, sigma(n) + if( n%3==0, -1 * sigma(n/3)) + if( n%4==0, -4 * sigma(n/4)) + if( n%12==0, +4 * sigma(n/12)))};

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, 3, p==3, 3^e, (p^(e+1) - 1) / (p - 1))))};

Formula

Expansion of (phi(q)^4 - phi(q^3)^4) / 8 = q * phi(q^3) * (chi(q) * psi(-q^3))^3 = q * psi(-q^3)^4 * (chi(q) * chi(q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

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

Euler transform of period 12 sequence [3, -3, 2, 0, 3, -6, 3, 0, 2, -3, 3, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A214456.

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

G.f.: x * Product_{k>0} (1 + (-x)^(3*k)) * (1 - x^(6*k))^4 / ( 1 + (-x)^k)^3.

a(n) = -(-1)^n * A124449(n). a(3*n) = 3*a(n). a(2*n) = a(4*n) = 3 * A121443(n). a(2*n + 1) = A185717(n).

From Amiram Eldar, Sep 12 2023: (Start)

Dirichlet g.f.: (1 - 1/2^(2*s-2)) * (1 - 1/3^s) * zeta(s-1) * zeta(s).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/18 = 0.548311... (A086463). (End)