A329958 Expansion of q^(-13/24) * eta(q^2)^3 * eta(q^3) * eta(q^6) / eta(q)^2 in powers of q.

1, 2, 2, 3, 3, 4, 4, 3, 5, 3, 6, 7, 4, 5, 4, 8, 6, 5, 7, 6, 7, 8, 7, 5, 8, 10, 9, 4, 7, 7, 9, 11, 8, 10, 5, 10, 12, 7, 10, 8, 10, 12, 4, 10, 8, 13, 15, 10, 9, 5, 15, 9, 12, 11, 10, 12, 10, 11, 11, 12, 15, 12, 6, 14, 8, 11, 17, 13, 12, 9, 16, 17, 8, 15, 10, 14

Offset

0,2

Links

Table of n, a(n) for n=0..75.

Formula

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

G.f.: Product_{k>=1} (1 + x^k)^2 * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)).

Convolution of A033762 and A080995. Convolution of A010054 and A121444.

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (3/2)^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329955.

Example

G.f. = 1 + 2*x + 2*x^2 + 3*x^3 + 3*x^4 + 4*x^5 + 4*x^6 + 3*x^7 + 5*x^8 + ...
G.f. = q^13 + 2*q^37 + 2*q^61 + 3*q^85 + 3*q^109 + 4*q^133 + 4*q^157 + ...

Mathematica

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

Prog

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

Crossrefs

Cf. A010054, A033762, A080995, A121444, A329955.

Sequence in context: A234475 A339082 A329907 * A309969 A036041 A252759

Adjacent sequences: A329955 A329956 A329957 * A329959 A329960 A329961

Keyword

nonn

Author

Michael Somos, Nov 26 2019

Status

approved