A226132 Expansion of - c(-q) * c(q^2) / 9 in powers of q where c() is a cubic AGM theta function.

1, -1, 3, -1, 6, -3, 8, -1, 9, -6, 12, -3, 14, -8, 18, -1, 18, -9, 20, -6, 24, -12, 24, -3, 31, -14, 27, -8, 30, -18, 32, -1, 36, -18, 48, -9, 38, -20, 42, -6, 42, -24, 44, -12, 54, -24, 48, -3, 57, -31, 54, -14, 54, -27, 72, -8, 60, -30, 60, -18, 62, -32, 72

Offset

1,3

Comments

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

Number 91 of the 126 eta-quotients listed in Table 1 of Williams 2012.

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.

K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Formula

Expansion of (a(q) - a(q^2)) * (a(q^2) + 2 * a(q^4)) / 18 = c(q^2)^4 / (9 * c(q) * c(q^4)) = (b(-q) * b(q^2) - a(-q) * a(q^2)) / 9 in powers of q where a(), b(), c(q) are cubic AGM theta functions.

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

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

Euler transform of period 12 sequence [ -1, 3, 2, 2, -1, -6, -1, 2, 2, 3, -1, -4, ...].

Multiplicative with a(2^e) = -1 if e>0, a(3^e) = 3^e, a(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)) = 4/3 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226139.

G.f.: Sum_{k>0 not 3|k} x^k / (1 - (-x)^k)^2 = Sum_{k>0 not 2|k} k * x^k * (1 - x^k) / (1 + x^(3*k)).

G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(4*k)) * (1 - x^(3*k))^6 * (1 + x^(3*k))^9 / ((1 - x^(2*k))^4 * (1 + x^(6*k))^3).

a(2*n) = - A121443(n). a(2*n + 1) = A185717(n).

a(n) = -(-1)^n * A121443(n). Convolution of A113447 and A113973.

From Amiram Eldar, Sep 12 2023: (Start)

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

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/54 = 0.18277045... . (End)

Example

G.f. = q - q^2 + 3*q^3 - q^4 + 6*q^5 - 3*q^6 + 8*q^7 - q^8 + 9*q^9 - 6*q^10 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ If[ OddQ[d] && ! Divisible[ n/d, 3], -d (-1)^(n/d), 0], {d, Divisors[ n]}]];
a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (Which[ # == 2, -1, # == 3, #^#2, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3]^3 / EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, I q^(3/2)]^3 / EllipticTheta[ 2, 0, I q^(1/2)] / (4 (-1)^(1/4)), {q, 0, n}];

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, if(d%2 && (n/d)%3, -d * (-1)^(n/d))))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k= 1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -1, if( p==3, p^e, (p^(e+1) - 1) / (p - 1))))))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^12 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^12 + A)^3), n))};

Crossrefs

Cf. A113447, A113973, A121443, A185717, A226139.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A321194 A158822 A353924 * A121443 A008795 A165188

Adjacent sequences: A226129 A226130 A226131 * A226133 A226134 A226135

Keyword

sign,easy,mult

Author

Michael Somos, May 27 2013

Status

approved