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
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.
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)
Comments