A258108 - OEIS

%I #7 Oct 19 2017 22:40:03

%S 1,3,0,-3,6,0,-12,15,0,-30,36,0,-60,78,0,-117,150,0,-228,276,0,-420,

%T 504,0,-732,885,0,-1245,1488,0,-2088,2454,0,-3420,3996,0,-5460,6378,0,

%U -8583,9972,0,-13344,15378,0,-20448,23472,0,-30876,35379,0,-46116,52644

%N Expansion of b(-q) * b(q^6) / (b(q^3) * b(q^12)) in powers of q where b() is a cubic AGM theta function.

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A258108/b258108.txt">Table of n, a(n) for n = 0..1000</a>

%F Expansion of eta(q^2)^9 * eta(q^9) * eta(q^36) / (eta(q)^3 * eta(q^3)^2 * eta(q^4)^3 * eta(q^12)^2 * eta(q^18)) in powers of q.

%F Euler transform of period 36 sequence [ 3, -6, 5, -3, 3, -4, 3, -3, 4, -6, 3, 1, 3, -6, 5, -3, 3, -4, 3, -3, 5, -6, 3, 1, 3, -6, 4, -3, 3, -4, 3, -3, 5, -6, 3, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A164616.

%F a(3*n + 2) = 0. a(3*n + 1) = 3 * A132977(n). a(3*n) = -3 * A164617(n) unless n = 0.

%e G.f. = 1 + 3*q - 3*q^3 + 6*q^4 - 12*q^6 + 15*q^7 - 30*q^9 + 36*q^10 + ...

%t QP=QPochhammer; A258108[n_] :=SeriesCoefficient[(QP[x^2]^9*QP[x^9]*QP[x^36])/(QP[x]^3*QP[x^3]^2*QP[x^4]^3*QP[x^12]^2*QP[x^18]), {x, 0, n}]; Table[A258108[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 18 2017 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A)^3 * eta(x^12 + A)^2 * eta(x^18 + A)), n))};

%Y Cf. A132977, A164616, A164617.

%K sign

%O 0,2

%A _Michael Somos_, May 20 2015