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A227213 Expansion of (eta(q^5) * eta(q^10) / (eta(q) * eta(q^2)))^2 in powers of q. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A290682 A000147 A128902 * A319455 A060552 A274868

Adjacent sequences:  A227210 A227211 A227212 * A227214 A227215 A227216

KEYWORD

nonn

AUTHOR

Michael Somos, Sep 19 2013

STATUS

approved

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Last modified January 22 16:37 EST 2020. Contains 331152 sequences. (Running on oeis4.)