A281722 - OEIS

%I #30 Oct 19 2018 09:18:58

%S 1,3,-18,12,21,-36,36,24,-90,12,54,-72,84,42,-144,72,93,-108,36,60,

%T -252,96,108,-144,180,93,-252,12,168,-180,216,96,-378,144,162,-288,84,

%U 114,-360,168,270,-252,288,132,-504,72,216,-288,372,171,-558,216,294,-324

%N Expansion of r(q) * s(q) in powers of q where r(), s() are cubic AGM functions.

%C Cubic AGM theta functions: r(q) (see A004016), s(q) (A005928), t(q) (A005882).

%H Seiichi Manyama, <a href="/A281722/b281722.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from G. C. Greubel)

%F Convolution of the sequences A004016 and A005928.

%F The g.f. is the product of the g.f.'s for A004016 and A005928. - _N. J. A. Sloane_, Jan 30 2017

%F Expansion of eta(q)^3 * (eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3)^2 in powers of q.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A144614.

%F a(3*n + 2) = A096726(3*n + 2) - 27 * A033686(n). a(n) == A096726(n) (mod 27). - _Michael Somos_, Sep 04 2017

%e G.f. = 1 + 3*q - 18*q^2 + 12*q^3 + 21*q^4 - 36*q^5 + 36*q^6 + 24*q^7 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 (QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / QPochhammer[ q^3]^2, {q, 0, n}];

%o (PARI) {a(n) = if( n<0, 0, my(A = x * O(x^n)); polcoeff( eta(x + A)^3 * (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) / eta(x^3 + A)^2, n))};

%Y Cf. A004016, A005928, A033686, A096726, A144614.

%K sign

%O 0,2

%A _Michael Somos_, Jan 28 2017