A198956 q-expansion of modular form psi_0^4/t_{3B}.

0, 1, 9, 27, 73, 126, 243, 344, 585, 729, 1134, 1332, 1971, 2198, 3096, 3402, 4681, 4914, 6561, 6860, 9198, 9288, 11988, 12168, 15795, 15751, 19782, 19683, 25112, 24390, 30618, 29792, 37449, 35964, 44226, 43344, 53217, 50654, 61740, 59346, 73710, 68922, 83592

Offset

0,3

Comments

psi_0 is given in A004016, t_{3B} in A198955.

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..1000

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]

Formula

Expansion of a(q) * (c(q) / 3)^3 in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Aug 23 2012

Expansion of eta(q^3)^8 * (1 + 9 * (eta(q^9) / eta(q))^3) in powers of q. - Michael Somos, Aug 23 2012

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = (1/3) (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A215711. - Michael Somos, Aug 23 2012

Convolution of A004016 and A106402. - Michael Somos, Aug 23 2012

Conjecture: Multiplicative with a(3^e) = 3^e, a(p^e) = sigma_3(p^e) for prime p <> 3. - Andrew Howroyd, Aug 08 2018

Example

G.f. = q + 9*q^2 + 27*q^3 + 73*q^4 + 126*q^5 + 243*q^6 + 344*q^7 + 585*q^8 + 729*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3]^8 (1 + 9 q (QPochhammer[ q^9] / QPochhammer[ q])^3), {q, 0, n}]; (* Michael Somos, Dec 27 2014 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^8 * (1 + 9 * x * (eta(x^9 + A) / eta(x + A))^3), n))}; /* Michael Somos, Aug 23 2012 */
(Sage) ModularForms( Gamma0(3), 4, prec=43).1; # Michael Somos, May 23 2014
(Magma) Basis( ModularForms( Gamma0(3), 4), 43)[2]; /* Michael Somos, Dec 27 2014 */

Crossrefs

Cf. A001158, A004016, A106402, A198955, A215711.

Sequence in context: A337628 A129957 A328408 * A110205 A319085 A211531

Adjacent sequences: A198953 A198954 A198955 * A198957 A198958 A198959

Keyword

nonn

Author

N. J. A. Sloane, Nov 01 2011

Status

approved