A232343 Expansion of q^(-5/3) * c(q^2)^3 / (9 * c(q)) in powers of q where c() is a cubic AGM theta function.

1, -1, 2, 0, 3, -2, 4, 0, 5, -5, 8, 0, 7, -4, 8, 0, 9, -8, 10, 0, 14, -6, 12, 0, 16, -14, 14, 0, 15, -8, 20, 0, 17, -14, 18, 0, 19, -10, 24, 0, 26, -21, 22, 0, 23, -16, 28, 0, 25, -20, 32, 0, 32, -14, 28, 0, 29, -28, 30, 0, 38, -16, 32, 0, 33, -31, 40, 0, 40

Offset

0,3

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Formula

Expansion of q^(-5/3) * eta(q) * eta(q^6)^9 / (eta(q^2) * eta(q^3))^3 in powers of q.

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

a(n) = 1/6 * b(3*n + 5) where b() is multiplicative with b(2^e) = 2 - 2^e, b(3^e) = 0^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.

a(2*n) = A098098(n). a(4*n + 1) = - A033686(n). a(4*n + 3) = 0.

Example

G.f. = 1 - x + 2*x^2 + 3*x^4 - 2*x^5 + 4*x^6 + 5*x^8 - 5*x^9 + 8*x^10 + ...
G.f. = q^5 - q^8 + 2*q^11 + 3*q^17 - 2*q^20 + 4*q^23 + 5*q^29 - 5*q^32 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^6]^9 / (QPochhammer[ x^2] QPochhammer[ x^3])^3, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3] QPochhammer[ x^12])^3 / (QPochhammer[ -x] QPochhammer[ x^4]), {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[# == 2, 2 - 2^#2, # == 3, 1, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[3 n + 5]) / 6]; (* Michael Somos, Jul 09 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^9 / (eta(x^2 + A) * eta(x^3 + A))^3, n))};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 5; A = factor(n); 1/6 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 2 - 2^e, p==3, 0, (p^(e+1) - 1) / (p - 1))))};
(Magma) Basis( ModularForms( Gamma0(18), 2), 210) [6]; /* Michael Somos, Jul 09 2018 */

Crossrefs

Cf. A033686, A098098.

Sequence in context: A208435 A208457 A353335 * A140944 A057860 A092915

Adjacent sequences: A232340 A232341 A232342 * A232344 A232345 A232346

Keyword

sign

Author

Michael Somos, Nov 22 2013

Status

approved