OFFSET
0,2
COMMENTS
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Yaacov Kopeliovich, Modular equations of order p and Theta functions, arXiv:0705.3914 [math.CV], 2007, p. 9 (theorem 3.1).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function
FORMULA
Expansion of 1 - lambda(tau) = lambda( -1 / tau) in powers of q = exp(pi i tau).
Expansion of (eta(q^4) * eta(q)^2 / eta(q^2)^3)^8 in powers of q.
Expansion of (phi(-q) / phi(q))^4 = (phi(-q) / phi(-q^2))^8 = (phi(-q^2) / phi(q))^8 = (f(-q) / f(q))^8 = (chi(-q) / chi(q))^8 = (psi(-q) / psi(q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -16, 8, -16, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 * (1 - u)^2 - 16 * u * (1 - v).
G.f.: (Product_{k>0} (1 - x^(2*k - 1)) / (1 + x^(2*k - 1)))^8 = exp( -16 * Sum_{k>0} x^(2*k - 1) * sigma(2*k - 1) / (2*k - 1)).
Empirical : Sum_{n=1..infinity} (exp(-2*Pi)^(n-1)*a(n)) = -16+12*2^(1/2). - Simon Plouffe, Feb. 20, 2011.
Empirical : Sum_{n=1..infinity} exp(-Pi*sqrt(3))^(n-1)*(-1)^(n+1)*a(n) = 8 - 4*sqrt(3). - Simon Plouffe, Feb. 20, 2011.
G.f. A(x) satisfies A(x) = 1/A(-x). - Michael Somos, Oct 07 2024
G.f. A(q) satisfies 1 = (A(q)*A(q^3))^(1/4) + ((1 - A(q))*(1 - A(q^3)))^(1/4). See Kopeliovich link. - Paul D. Hanna, Oct 14 2024
EXAMPLE
G.f. A(q) = 1 - 16*q + 128*q^2 - 704*q^3 + 3072*q^4 - 11488*q^5 + 38400*q^6 + ...
RELATED SERIES.
(A(q)*A(q^3))^(1/4) = 1 - 4*x + 8*x^2 - 20*x^3 + 48*x^4 - 88*x^5 + 168*x^6 - 320*x^7 + 544*x^8 - 932*x^9 + ... + (-1)^n*A123861(n)*x^n + ...
where ((1 - A(q))*(1 - A(q^3)))^(1/4) = 1 - (A(q)*A(q^3))^(1/4).
MATHEMATICA
CoefficientList[(QPochhammer[q]/QPochhammer[-q])^8 + O[q]^30, q] (* Jean-François Alcover, Nov 05 2015 *)
eta[q_] := q^(1/24)*QPochhammer[q]; a[n_] := SeriesCoefficient[(eta[q^4]* eta[q]^2/eta[q^2]^3)^8, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 18 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A) / eta(x^2 + A)^3)^8, n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( exp(-16 * sum(k=1, (n+1)\2, sigma(2*k-1) / (2*k-1) * x^(2*k-1), x * O(x^n))), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 20 2007
STATUS
approved