cp's OEIS Frontend

A128486 Expansion of ((b(q)*c(q))^3 - 8*(b(q^2)*c(q^2))^3) / 27 in powers of q where b(), c() are cubic AGM theta functions.

%I A128486 #7 Sep 08 2022 08:45:30
%S A128486 1,-14,9,52,6,-126,-40,136,81,-84,-564,468,638,560,54,-2480,882,-1134,
%T A128486 -556,312,-360,7896,-840,1224,-3089,-8932,729,-2080,4638,-756,4400,
%U A128486 10528,-5076,-12348,-240,4212,-2410,7784,5742,816,-6870,5040,9644,-29328,486,11760,-18672,-22320
%N A128486 Expansion of ((b(q)*c(q))^3 - 8*(b(q^2)*c(q^2))^3) / 27 in powers of q where b(), c() are cubic AGM theta functions.
%C A128486 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%F A128486 Expansion of b(q) * b(q^2) * c(q) * c(q^2) * (b(q) * b(q^2) - c(q) * c(q^2)) / 9 in powers of q where b(), c() are cubic AGM theta functions.
%F A128486 Expansion of (eta(q) * eta(q^3))^6 - 8*(eta(q^2) * eta(q^6))^6 in powers of q.
%F A128486 Expansion of eta(q) * eta(q^2) * eta(q^3) * eta(q^6) * ((eta(q) * eta(q^2))^4 - 9*(eta(q^3) * eta(q^6))^4) in powers of q.
%F A128486 G.f.: x * Product_{k>0} (1 - x^k)^6 * (1 - x^(3*k))^6 - 8 * x^2 * Product_{k>0} (1 - x^(2*k))^6 * (1 - x^(6*k))^6.
%F A128486 G.f. is a period 1 Fourier series which satisfies f(-1 / (6*t)) = -216 (t/i)^6 f(t) where q = exp(2 Pi i t).
%F A128486 a(2*n) = A007332(2*n) - 8 * A007332(n). a(2*n + 1) = A007332(2*n + 1).
%e A128486 G.f. = q - 14*q^2 + 9*q^3 + 52*q^4 + 6*q^5 - 126*q^6 - 40*q^7 + 136*q^8 + ...
%o A128486 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^6 - 8*x * (eta(x^2 + A) * eta(x^6+A))^6, n))};
%o A128486 (PARI) {a(n) = my(A, A1, A2); if( n<1, 0, n--; A = x * O(x^n); A1 = eta(x + A) * eta(x^2 + A); A2 = eta(x^3 + A) * eta(x^6 + A); polcoeff( A1^5*A2 - 9*x * A1*A2^5, n))};
%o A128486 (Magma) A := Basis( CuspForms( Gamma1(6), 6), 49); A[1] - 14*A[2] + 9*A[3]; /* _Michael Somos_, Feb 19 2015 */
%Y A128486 Cf. A007332.
%K A128486 sign,mult
%O A128486 1,2
%A A128486 _Michael Somos_, Mar 04 2007