A329955 Expansion of eta(q) * eta(q^2) * eta(q^3)^3 / eta(q^6)^2 in powers of q.

1, -1, -2, -2, 3, 8, 0, -2, -10, -4, 2, 4, 10, -8, -4, 0, 7, 12, 4, -2, -16, -16, 4, 8, 0, -7, -4, -2, 10, 24, 8, -2, -26, 0, 2, 8, 12, -16, -8, -8, 10, 12, 0, -6, -20, -16, 4, 8, 26, -7, -10, 0, 16, 40, 0, -4, -20, -24, 6, 4, 0, -16, -12, -8, 15, 24, 8, -6

Offset

0,3

Links

Table of n, a(n) for n=0..67.

Formula

Euler transform of period 6 sequence [-1, -2, -4, -2, -1, -3, ...].

G.f.: Product_{k>=1} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) / (1 + x^(3*k))^2.

Convolution of A030206 and A195848.

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 1990656^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329958.

a(3*n) = A224822(n). a(3*n + 1) = -A329956(n). a(3*n + 2) = -2*A329957(n). a(6*n) = A028967(n).

Example

G.f. = 1 - x - 2*x^2 - 2*x^3 + 3*x^4 + 8*x^5 - 2*x^7 - 10*x^8 - 4*x^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2] QPochhammer[ x^3]^3 / QPochhammer[ x^6]^2, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A)^3 / eta(x^6 + A)^2, n))};

Crossrefs

Cf. A028967, A030206, A195848, A224822, A329956, A329957.

Sequence in context: A143673 A089543 A139073 * A099870 A221877 A110985

Adjacent sequences: A329952 A329953 A329954 * A329956 A329957 A329958

Keyword

sign

Author

Michael Somos, Nov 26 2019

Status

approved