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A136747 Expansion of a(q)^2 * (b(q) * c(q) / 3)^3 in powers of q where a(), b(), c() are cubic AGM theta functions.

%I A136747 #26 Aug 15 2025 03:15:51
%S A136747 1,6,-27,-92,390,-162,-64,-1320,729,2340,-948,2484,-5098,-384,-10530,
%T A136747 3856,28386,4374,-8620,-35880,1728,-5688,-15288,35640,73975,-30588,
%U A136747 -19683,5888,36510,-63180,-276808,192096,25596,170316,-24960,-67068,268526,-51720,137646
%N A136747 Expansion of a(q)^2 * (b(q) * c(q) / 3)^3 in powers of q where a(), b(), c() are cubic AGM theta functions.
%C A136747 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A136747 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, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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.] See page 30, line -3.
%H A136747 LMFDB, <a href="https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/3/8/a/a/">Newform orbit 3.8.a.a</a>.
%F A136747 Expansion of (eta(q) * eta(q^3))^6 * ((eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3))^2 in powers of q.
%F A136747 a(n) is multiplicative with a(3^e) = (-27)^e, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) unless p = 3.
%F A136747 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^4*w + 512*u^3*w^2 + 131072*u^2*w^3 + 16777216*u*w^4 - 24*u^3*v*w - 9216*u^2*v*w^2 - 1572864*u*v*w^3 + 288*u^2*v^2*w + 73728*u*v^2*w^2 - u^2*v^3 - 1984*w*v^3*u - 65536*w^2*v^3 + 12*v^4*u + 3072*w*v^4 - 36*v^5.
%F A136747 G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 81 (t/i)^8 f(t) where q = exp(2 Pi i t).
%F A136747 G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(3*k)))^6 * (Sum_{j,k in Z} x^(j*j + j*k + k*k))^2.
%F A136747 Convolution of A007332 and A008653.
%e A136747 G.f. = q + 6*q^2 - 27*q^3 - 92*q^4 + 390*q^5 - 162*q^6 - 64*q^7 - 1320*q^8 + ...
%t A136747 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^3])^6 ((QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / QPochhammer[ q^3])^2, {q, 0, n}]; (* _Michael Somos_, May 28 2013 *)
%o A136747 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^6 * sum(k=1, n, 12 * (sigma(3*k) - 3 * sigma(k)) * x^k, 1 + A), n))};
%o A136747 (Sage) CuspForms( Gamma0(3), 8, prec=40).0; # _Michael Somos_, May 28 2013
%o A136747 (Magma) Basis( CuspForms( Gamma0(3), 8), 40)[1]; /* _Michael Somos_, Oct 12 2015 */
%Y A136747 Cf. A007332, A008653.
%K A136747 sign,mult
%O A136747 1,2
%A A136747 _Michael Somos_, Jan 20 2008