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A121589 Series expansion of (eta(q^9) / eta(q))^3 in powers of q. 6
1, 3, 9, 22, 51, 108, 221, 429, 810, 1476, 2631, 4572, 7802, 13056, 21519, 34918, 55935, 88452, 138332, 213990, 327852, 497592, 748833, 1117692, 1655719, 2434938, 3556791, 5161808, 7445631, 10677096, 15226658, 21599469, 30485268, 42817788 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

LINKS

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

Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

FORMULA

Euler transform of period 9 sequence [3, 3, 3, 3, 3, 3, 3, 3, 0, ...].

G.f.: x * (Product_{k>0} (1 - x^(9*k) / (1 - x^k))^3.

Expansion of c(q^3) / (3 * b(q)) = (c(q) / (3 * b(q^3))^3 in powers of q where b(), c() are cubic AGM functions.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (u^2 - v) - 2 * u * v * ( 3 * (u + v) + 13 * u * v ).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v * (1 + 9 * v + 27 * v^2) * (1 + 9 * u + 27 * u^2).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 * u2 - (1 + 3 * (u1 + u2)) * (u3 + u6 + 9 * u3 * u6).

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/27) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A131986.

a(n) ~ exp(4*Pi*sqrt(n)/3) / (27 * sqrt(6) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

a(1) = 1, a(n) = (3/(n-1))*Sum_{k=1..n-1} A116607(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017

Convolution inverse of A131986. Convolution cube of A104502. - Michael Somos, Nov 02 2017

EXAMPLE

G.f. = q + 3*q^2 + 9*q^3 + 22*q^4 + 51*q^5 + 108*q^6 + 221*q^7 + 429*q^8 + ...

MAPLE

N:= 100: # to get a(1)..a(N)

S:= series(q*Product(1-q^(9*k), k=1..N/9)/Product((1-q^k)^3, k=1..N), q, N+1):

seq(coeff(S, q, n), n=1..N); # Robert Israel, Nov 02 2017

MATHEMATICA

nmax = 40; CoefficientList[Series[Product[((1-x^(9*k))/(1-x^k))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)

QP = QPochhammer; s = (QP[q^9]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 30 2015, adapted from PARI *)

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^9] / QPochhammer[ q])^3, {q, 0, n}]; (* Michael Somos, Nov 02 2017 *)

PROG

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^9 + A) / eta(x + A))^3, n))};

CROSSREFS

Cf. A104502, A131986.

Sequence in context: A000711 A278668 A160526 * A227454 A000716 A001628

Adjacent sequences:  A121586 A121587 A121588 * A121590 A121591 A121592

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 09 2006

EXTENSIONS

Second formula corrected by Vaclav Kotesovec, Sep 07 2015

STATUS

approved

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Last modified May 26 18:08 EDT 2020. Contains 334630 sequences. (Running on oeis4.)