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 A122830 Expansion of c(q) * c(q^6) / c(q^2)^2 in powers of q where c() is a cubic AGM theta function. 3

%I

%S 1,1,0,-2,-3,0,5,7,0,-12,-15,0,26,32,0,-50,-63,0,92,114,0,-168,-201,0,

%T 295,350,0,-496,-591,0,818,967,0,-1332,-1554,0,2126,2468,0,-3324,

%U -3855,0,5126,5916,0,-7824,-8970,0,11793,13471,0,-17548,-20007,0,25857,29384,0,-37788,-42771,0

%N Expansion of c(q) * c(q^6) / c(q^2)^2 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).

%H G. C. Greubel, <a href="/A122830/b122830.txt">Table of n, a(n) for n = 1..1000</a>

%F Expansion of eta(q^2)^2 * eta(q^3)^3 * eta(q^18)^3 / (eta(q) * eta(q^6)^7) in powers of q.

%F Euler transform of period 18 sequence [ 1, -1, -2, -1, 1, 3, 1, -1, -2, -1, 1, 3, 1, -1, -2, -1, 1, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = 4*v - 4*u^2 + 4*v^2 + 2*w^2 + 8*u*v + 8*v*w + 18*u*v*w + 3*u*w^2 - 12*u^2*w - 12*u^2*v + 6*v^2*w - 3*v^3 - 9*u^2*w^2 - 18*u^2*v*w - 9*u*v^2*w - 9*u^2*v^2 - 9*v^3*w.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = 2*u6*u1 - 2*u3*u2 + u6*u2^2 - 3*u6*u3*u2 - 3*u3^2*u2 - 4*u3*u2^2 - 3*u3^2*u2^2 + 6*u3^2*u1 - 4*u3*u2*u1 + 4*u6*u2*u1 + u6*u1^2 + 2*u3*u1^2 + 6*u3^2*u1^2 + 3*u6*u3*u1^2 - 6*u6*u3*u2^2 + 6*u3^2*u2*u1 - 6*u6*u3*u2*u1.

%F Convolution inverse is A182033. - _Michael Somos_, Feb 19 2015

%F a(3*n) = 0. - _Michael Somos_, Feb 19 2015

%e G.f. = q + q^2 - 2*q^4 - 3*q^5 + 5*q^7 + 7*q^8 - 12*q^10 - 15*q^11 + ...

%t eta[x_] := x^(1/24)*QPochhammer[x]; A122830[n_] := SeriesCoefficient[

%t eta[q^2]^2* eta[q^3]^3*eta[q^18]^3/(eta[q]*eta[q^6]^7 ), {q, 0, n}]; Table[A122830[n], {n, 0, 50}] (* _G. C. Greubel_, Aug 11 2017 *)

%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^18 + A)^3 / (eta(x + A) * eta(x^6 + A)^7), n))};

%Y Cf. A182033.

%K sign

%O 1,4

%A _Michael Somos_, Sep 12 2006

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)