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%I #7 Sep 08 2022 08:46:08
%S 1,1,2,0,2,2,3,2,1,4,5,4,6,1,8,4,7,4,2,6,8,6,12,0,10,7,14,8,2,8,17,8,
%T 14,2,16,12,16,10,3,8,18,10,20,2,18,10,23,16,1,14,24,16,20,4,30,16,22,
%U 16,5,16,24,18,30,4,28,14,32,18,6,20,33,16,26,1
%N Expansion of c(q) * c(q^5) / 9 in powers of q where c() is a cubic AGM theta function.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%F Expansion of (eta(q^3) * eta(q^15))^3 / (eta(q) * eta(q^5)) in powers of q.
%F Euler transform of period 15 sequence [ 1, 1, -2, 1, 2, -2, 1, 1, -2, 2, 1, -2, 1, 1, -4, ...].
%F a(5*n) = a(n) for all n in Z.
%F Given g.f. A = A0 + A1 + A2 + A3 + A4 is the 5-section, then 0 = A4*A3^2 - A4^2*A2 + A2^2*A1 - A3*A1^2 - 6*A3*A2*A0 + 6*A4*A1*A0.
%e G.f. = q^2 + q^3 + 2*q^4 + 2*q^6 + 2*q^7 + 3*q^8 + 2*q^9 + q^10 + 4*q^11 + 5*q^12 + ...
%o (PARI) {a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^15 + A))^3 / (eta(x + A) * eta(x^5 + A)), n))};
%o (Magma) A := Basis( ModularForms( Gamma0(15), 2), 61); A[3] + A[4];
%K nonn
%O 2,3
%A _Michael Somos_, Nov 11 2014