A227213 Expansion of (eta(q^5) * eta(q^10) / (eta(q) * eta(q^2)))^2 in powers of q.

1, 2, 7, 14, 35, 64, 136, 238, 457, 770, 1377, 2248, 3822, 6072, 9920, 15406, 24386, 37114, 57240, 85590, 129152, 190104, 281542, 408616, 595425, 853244, 1225705, 1736304, 2462830, 3452240, 4841442, 6721262, 9329664, 12837572, 17653935, 24092998, 32850206

Offset

1,2

Comments

Number 9 of the 14 eta-quotients listed in Table 2 of Moy 2013.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Richard Moy, Congruences among power series coefficients of modular forms, arXiv:1309.4320 [math.NT], 2013.

Formula

Euler transform of period 10 sequence [ 2, 4, 2, 4, 0, 4, 2, 4, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 1/25 * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132041.

G.f.: x * (Product_{k>0} ( (1 - x^(5*k)) * (1 - x^(10*k))) / ((1 - x^k) * (1 - x^(2*k))))^2. [corrected by Vaclav Kotesovec, Sep 08 2015]

Convolution inverse of A132041.

a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(3/4) * 5^(9/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

Example

G.f. = q + 2*q^2 + 7*q^3 + 14*q^4 + 35*q^5 + 64*q^6 + 136*q^7 + 238*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^5] QPochhammer[ q^10] / (QPochhammer[ q] QPochhammer[ q^2]))^2, {q, 0, n}]; (* Michael Somos, Jan 10 2015 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(5*k)) * (1 - x^(10*k)) / ((1 - x^k) * (1 - x^(2*k))))^2, {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^5 + A) * eta(x^10 + A) / (eta(x + A) * eta(x^2 + A)))^2, n))};

Crossrefs

Cf. A132041.

Sequence in context: A000147 A334069 A128902 * A319455 A060552 A274868

Adjacent sequences: A227210 A227211 A227212 * A227214 A227215 A227216

Keyword

nonn

Author

Michael Somos, Sep 19 2013

Status

approved