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

%I #28 Sep 08 2022 08:46:00

%S 0,1,9,27,73,126,243,344,585,729,1134,1332,1971,2198,3096,3402,4681,

%T 4914,6561,6860,9198,9288,11988,12168,15795,15751,19782,19683,25112,

%U 24390,30618,29792,37449,35964,44226,43344,53217,50654,61740,59346,73710,68922,83592

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

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

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

%H G. C. Greubel, <a href="/A198956/b198956.txt">Table of n, a(n) for n = 0..1000</a>

%H Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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.]

%F 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

%F Expansion of eta(q^3)^8 * (1 + 9 * (eta(q^9) / eta(q))^3) in powers of q. - _Michael Somos_, Aug 23 2012

%F 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

%F Convolution of A004016 and A106402. - _Michael Somos_, Aug 23 2012

%F Conjecture: Multiplicative with a(3^e) = 3^e, a(p^e) = sigma_3(p^e) for prime p <> 3. - _Andrew Howroyd_, Aug 08 2018

%e 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 + ...

%t 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 *)

%o (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 */

%o (Sage) ModularForms( Gamma0(3), 4, prec=43).1;# _Michael Somos_, May 23 2014

%o (Magma) Basis( ModularForms( Gamma0(3), 4), 43)[2]; /* _Michael Somos_, Dec 27 2014 */

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

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Nov 01 2011