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A128638 Expansion of q * (psi(q^3)^3 / psi(q)) / (phi(-q)^3 / phi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions. 8
1, 5, 19, 61, 174, 455, 1112, 2573, 5689, 12102, 24900, 49759, 96902, 184408, 343722, 628717, 1130418, 2000669, 3489788, 6005910, 10207688, 17147892, 28494120, 46865519, 76342903, 123236446, 197233723, 313106264, 493231830, 771301986, 1197743552, 1847606573 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (1/3) * (c(q^2)^2 / c(q)) / (b(q)^2 / b(q^2)) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (eta(q^6) / eta(q))^5 * eta(q^2) / eta(q^3) in powers of q.
Euler transform of period 6 sequence [ 5, 4, 6, 4, 5, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1 + 8*u) * (1 + 9*v) - (u-v)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/72) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128632.
G.f.: x * Product_{k>0} ((1 - x^(6*k)) / (1-x^k))^5 * ((1 - x^(2*k)) / (1 - x^(3*k))).
8 * a(n) = A128639(n) unless n = 0. Convolution inverse of A128632.
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (72 * 2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
EXAMPLE
G.f. = q + 5*q^2 + 19*q^3 + 61*q^4 + 174*q^5 + 455*q^6 + 1112*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^6] / QPochhammer[ q])^5 (QPochhammer[ q^2] / QPochhammer[ q^3]), {q, 0, n}]; (* Michael Somos, Jun 08 2015 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(6*k)) / (1-x^k))^5 * ((1 - x^(2*k)) / (1 - x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 08 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^6 + A) / eta(x + A))^5 * eta(x^2 + A) / eta(x^3 + A), n))};
CROSSREFS
Sequence in context: A135266 A124123 A189714 * A036630 A338229 A102841
KEYWORD
nonn
AUTHOR
Michael Somos, Mar 16 2007
EXTENSIONS
Edited by N. J. A. Sloane, Apr 01 2008
STATUS
approved

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Last modified July 17 21:15 EDT 2024. Contains 374377 sequences. (Running on oeis4.)