cp's OEIS Frontend

A279005 Expansion of 3 * q * b(q^9)^3 / c(q^3) in powers of q^3 where b(), c() are cubic AGM theta functions.

%I A279005 #16 Aug 04 2018 12:17:13
%S A279005 1,-1,-1,-6,6,7,9,-8,-15,13,-19,-4,-49,57,61,32,-14,-75,47,-98,-23,
%T A279005 -124,130,152,116,-37,-182,96,-168,0,-335,352,342,196,-117,-390,230,
%U A279005 -440,-107,-600,637,671,480,-184,-704,469,-727,-112,-1235,1241,1291,722,-341
%N A279005 Expansion of 3 * q * b(q^9)^3 / c(q^3) in powers of q^3 where b(), c() are cubic AGM theta functions.
%C A279005 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A279005 G. C. Greubel, <a href="/A279005/b279005.txt">Table of n, a(n) for n = 0..2500</a>
%H A279005 Amanda Clemm, <a href="http://www.mdpi.com/2227-7390/4/1/5">Modular Forms and Weierstrass Mock Modular Forms</a>, Mathematics, volume 4, issue 1, (2016)
%F A279005 Expansion of q * eta(q^3) * eta(q^9)^6 / eta(q^27)^3 in powers of q^3.
%F A279005 Euler transform of period 9 sequence [ -1, -1, -7, -1, -1, -7, -1, -1, -4, ...].
%F A279005 a(5*n + 2) / a(2) == A030206(n) (mod 5). a(125*n + 42) / a(42) == A030206(n) (mod 25). [Amanda Clemm, 2016]
%e A279005 G.f. = 1 - x - x^2 - 6*x^3 + 6*x^4 + 7*x^5 + 9*x^6 - 8*x^7 - 15*x^8 + ...
%e A279005 G.f. = q^-1 - q^2 - q^5 - 6*q^8 + 6*q^11 + 7*q^14 + 9*q^17 - 8*q^20 + ...
%t A279005 a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^3]^6 / QPochhammer[ x^9]^3, {x, 0, n}];
%o A279005 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A)^6 / eta(x^9 + A)^3, n))};
%Y A279005 Cf. A030206.
%K A279005 sign
%O A279005 0,4
%A A279005 _Michael Somos_, Dec 10 2016