A116418 Expansion of q^(-1/3) * b(q) * c(q) * b(q^2) / 3 in powers of q where b(), c() are cubic AGM theta functions.

1, -2, -4, 6, 8, 4, -16, -24, 7, 8, 44, 18, -34, -12, -40, 24, -33, -16, 72, -6, 50, -8, 8, -24, -16, 32, -76, -66, -54, 48, -32, 120, 176, -14, -28, -54, 56, -16, -72, -48, -167, -88, 92, 48, 64, -36, 152, 72, 18, 68, -148, 96, -82, 24, 56, -168, -105, 80, -28, -18, -232, -48, 216, -96, 206, 66, 20, -42, 198, 32

Offset

0,2

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

LMFDB, Newform 18.3.17.a.

Formula

Expansion of q^(-1/3) * b(q) * c(q) * b(q^2) / 3 in powers of q where b(), c() are cubic AGM theta functions.

Expansion of q^(-1/3) * eta(q)^2 * eta(q^2)^3 * eta(q^3)^2 / eta(q^6) in powers of q.

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

Expansion of a newform level 18 weight 3 and character chi_18(17,.).

a(n) = b(3*n + 1) where b() is multiplicative with b(p^e) = b(p) * b(p^(e-1)) - Kronecker(-12, p) * p^2 * b(p^(e-2)). - Michael Somos, Jan 08 2017

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 3888^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208384.

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

a(n) = A208384(3*n + 1) = A208385(3*n + 1).

a(2*n + 1) = -2 * A122407(n). - Michael Somos, Mar 30 2015

a(4*n + 1) = -2 * a(n). a(4*n + 3) = 6 * A280822(n). - Michael Somos, Jan 08 2017

Example

G.f. = 1 - 2*x - 4*x^2 + 6*x^3 + 8*x^4 + 4*x^5 - 16*x^6 - 24*x^7 + 7*x^8 + ...
G.f. = q - 2*q^4 - 4*q^7 + 6*q^10 + 8*q^13 + 4*q^16 - 16*q^19 - 24*q^22 +...

Mathematica

s[n_] := Series[Product[(1-x^k)^2*(1-x^(2*k))^3*(1-x^(3*k))^2/(1-x^(6*k)), {k, 1, n}], {x, 0, n}] // Normal; a[k_] := SeriesCoefficient[s[n], {x, 0, k}]; a[0]=1; Table[a[n], {n, 0, 69}] (* Jean-François Alcover, Feb 04 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^2]^3 QPochhammer[ x^3]^2 / QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Mar 30 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A)^3 * eta(x^3 + A)^2 / eta(x^6 + A), n))};
(Magma) A := Basis( ModularForms( Gamma1(18), 3), 210); A[2] - 2*A[5] - 4*A[8] + 6*A[11] + 8*A[14] + 4*A[17] - 16*A[20] - 24*A[23]; /* Michael Somos, Mar 30 2015 */
(Magma) A := Basis( CuspForms( Gamma1(18), 3), 210); A[1] -2*A[4] - 4*A[7] + 6*A[10]; /* Michael Somos, Jan 08 2017 */

Crossrefs

Cf. A122407, A208384, A208385, A280822.

Sequence in context: A304090 A218454 A115425 * A366637 A360700 A122640

Adjacent sequences: A116415 A116416 A116417 * A116419 A116420 A116421

Keyword

sign

Author

Michael Somos, Feb 13 2006

Status

approved