A123647 Expansion of (eta(q^4) * eta(q^12) / (eta(q) * eta(q^3)))^2 in powers of q.

1, 2, 5, 12, 22, 42, 80, 136, 233, 396, 636, 1020, 1622, 2496, 3822, 5808, 8642, 12786, 18788, 27208, 39184, 56088, 79432, 111912, 156823, 217964, 301517, 415104, 567758, 773244, 1048616, 1414432, 1900524, 2543940, 3389792, 4501164, 5956430

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

Euler transform of period 12 sequence [ 2, 2, 4, 0, 2, 4, 2, 0, 4, 2, 2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v * (1 + 4*u) * (1 + 4*v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/16) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187196. - Michael Somos, Sep 02 2015

Convolution inverse of A187196. - Michael Somos, Sep 02 2015

a(n) ~ exp(2*Pi*sqrt(n/3)) / (32 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 08 2015

Example

G.f. = x + 2*x^2 + 5*x^3 + 12*x^4 + 22*x^5 + 42*x^6 + 80*x^7 + 136*x^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] QPochhammer[ q^12] / (QPochhammer[ q^] QPochhammer[ q^3]))^2, {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^4 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^3 + A)))^2, n))};

Crossrefs

Cf. A187196.

Sequence in context: A026035 A215183 A086734 * A166249 A326762 A116711

Adjacent sequences: A123644 A123645 A123646 * A123648 A123649 A123650

Keyword

nonn

Author

Michael Somos, Oct 04 2006

Status

approved