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A131943 Expansion of b(q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

%I A131943 #16 Sep 08 2022 08:45:31
%S A131943 1,-3,-3,15,-3,-18,15,-24,-3,69,-18,-36,15,-42,-24,90,-3,-54,69,-60,
%T A131943 -18,120,-36,-72,15,-93,-42,231,-24,-90,90,-96,-3,180,-54,-144,69,
%U A131943 -114,-60,210,-18,-126,120,-132,-36,414,-72,-144,15,-171,-93,270,-42,-162
%N A131943 Expansion of b(q) * b(q^2) in powers of q where b() is a cubic AGM theta function.
%C A131943 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D A131943 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.65).
%F A131943 Expansion of eta(q)^3 * eta(q^2)^3 / (eta(q^3) * eta(q^6)) in powers of q.
%F A131943 Euler transform of period 6 sequence [ -3, -6, -2, -6, -3, -4, ...].
%F A131943 a(n) = -3 * b(n) where b() is multiplicative with b(2^e) = 1, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3. - _Michael Somos_, Nov 21 2013
%F A131943 G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 54 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121443.
%F A131943 G.f.: Product_{k>0} ((1 - x^k) * (1 - x^(2*k)))^3 / ((1 - x^(3*k)) * (1 - x^(6*k))).
%F A131943 G.f.: 1 - 3 * (Sum_{k>0} (6*k - 1) * x^(6*k - 1) / (1 - x^(6*k - 1)) - 2*(6*k - 5) * x^(6*k - 3) / (1 - x^(6*k - 3)) + (6*k - 5) * x^(6*k - 5) / (1 -x^(6*k - 5))).
%F A131943 a(n) = a(2*n). a(n) = -3 * A131944(n) unless n=0. a(3^n) = 3 * A168611(n+1). a(2*n + 1) = -3 * A134077(n). - _Michael Somos_, Nov 21 2013
%e A131943 G.f. = 1 - 3*q - 3*q^2 + 15*q^3 - 3*q^4 - 18*q^5 + 15*q^6 - 24*q^7 - 3*q^8 +...
%t A131943 a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 QPochhammer[ q^2]^3 / (QPochhammer[ q^3] QPochhammer[ q^6]), {q, 0, n}]; (* _Michael Somos_, Nov 21 2013 *)
%t A131943 a[ n_] := If[ n < 1, Boole[n == 0], -3 Sum[ d {0, 1, 0, -2, 0, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* _Michael Somos_, Nov 11 2015 *)
%o A131943 (PARI) {a(n) = if( n<1, n==0, -3 * sumdiv(n, d, d * ((d%6==1) + (d%6==5) - 2 * (d%6==3))))};
%o A131943 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A))^3 / (eta(x^3 + A) * eta(x^6 + A)), n))};
%o A131943 (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -3 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 4 - 3^(e+1), (p^(e+1) - 1) / (p - 1) )))}; /* _Michael Somos_, Nov 21 2013 */
%o A131943 (Magma) A := Basis( ModularForms( Gamma0(6), 2), 54); A[1] - 3*A[2] - 3*A[3]; /* _Michael Somos_, Aug 30 2014 */
%o A131943 (Sage) A = ModularForms( Gamma0(6), 2, prec=54) . basis();  A[0] - 3*A[1] - 3*A[2]; # _Michael Somos_, Nov 21 2013
%Y A131943 Cf. A121443, A131944, A134077, A168611.
%K A131943 sign
%O A131943 0,2
%A A131943 _Michael Somos_, Jul 30 2007