The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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-François 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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