A128663 Expansion of eta(q^3) * eta(q^33) / ( eta(q)* eta(q^11)) in powers of q.

1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, 28, 37, 46, 59, 74, 94, 115, 144, 176, 218, 265, 326, 393, 479, 574, 695, 830, 996, 1184, 1414, 1673, 1988, 2344, 2770, 3254, 3828, 4482, 5252, 6126, 7153, 8318, 9678, 11222, 13018, 15050, 17405, 20068, 23145, 26621

Offset

1,3

Links

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

Formula

Euler transform of period 33 sequence [ 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (1 + 3 * u*v) - (u+v) * (u^2 - 3 * u*v + v^2).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = v - u^3 + 3 * u*v * (2*u + (1+v) * (1 + 3*u*v)).

G.f. is a period 1 Fourier series which satisfies f(-1 / (33 t)) = 1/3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226009.

G.f.: x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(33*k)) / ( (1 - x^k) * (1 - x^(11*k))).

Convolution inverse of A226009.

a(n) ~ exp(4*Pi*sqrt(n/33)) / (sqrt(2) * 3^(5/4) * 11^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

Example

q + q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 5*q^6 + 7*q^7 + 9*q^8 + 13*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3] QPochhammer[ q^33] / (QPochhammer[ q] QPochhammer[ q^11]), {q, 0, n}]
nmax = 40; Rest[CoefficientList[Series[x * Product[(1 - x^(3*k)) * (1 - x^(33*k)) / ( (1 - x^k) * (1 - x^(11*k))), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 08 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^33 + A) / (eta(x + A) * eta(x^11 + A)), n))}

Crossrefs

Cf. A226009.

Sequence in context: A058661 A094362 A000726 * A206557 A240508 A174068

Adjacent sequences: A128660 A128661 A128662 * A128664 A128665 A128666

Keyword

nonn

Author

Michael Somos, Mar 19 2007

Status

approved