A329651 Expansion of x * (psi(x^6) / psi(-x^3))^3 * phi(-x)^5 / psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

0, 1, -9, 31, -45, 6, 45, 8, -117, 121, -54, 12, 9, 14, -72, 186, -261, 18, 207, 20, -270, 248, -108, 24, -63, 31, -126, 391, -360, 30, 270, 32, -549, 372, -162, 48, 171, 38, -180, 434, -702, 42, 360, 44, -540, 726, -216, 48, -207, 57, -279, 558, -630, 54, 693

Offset

0,3

Comments

Number 105 of the 126 eta-quotients listed in Table 1 of Williams 2012.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 144 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A328788.

Links

Table of n, a(n) for n=0..54.

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

Euler transform of period 12 sequence [-9, -5, -6, -4, -9, -2, -9, -4, -6, -5, -9, -4, ...].

Expansion of x * phi(-x)^5 / psi(-x) * (f(-x^12) / f(-x^3))^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Expansion of eta(q)^9 * eta(q^12)^3 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^4)) in powers of q.

a(n) = s(n) - 12*s(n/2) + 27*s(n/3) - 16*s(n/4) if n>0 where s(x) = sum of divisors of x for integer x else 0.

a(n) = -(-1)^n * A133739(n). a(4*n + 2) = -9 * A134077(n). a(6*n + 5) = 6 * A098098(n).

Example

G.f. = x - 9*x^2 + 31*x^3 - 45*x^4 + 6*x^5 + 45*x^6 + 8*x^7 - 117*x^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/2 (EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, Pi/4, x^(3/2)])^3 EllipticTheta[ 4, 0, x]^5 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}] // PowerExpand;

Prog

(PARI) {a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, 0, s(n) - 12*s(n/2) + 27*s(n/3) - 16*s(n/4))};
(PARI) {a(n) = my(A); if( n < 1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^9 * eta(x^12 + A)^3 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^4 + A)), n))};
(Magma) A := Basis( ModularForms( Gamma0(12), 2), 52); A[2] - 9*A[3] + 31*A[4] - 45*A[5];

Crossrefs

Cf. A000203, A098098, A133739, A134077, A328788.

Sequence in context: A298587 A054310 A072887 * A133739 A266397 A288419

Adjacent sequences: A329648 A329649 A329650 * A329652 A329653 A329654

Keyword

sign

Author

Michael Somos, Nov 18 2019

Status

approved