%I #21 Aug 06 2024 20:51:30
%S 0,1,0,0,-1,0,-1,-2,0,1,1,-2,2,0,2,-1,3,4,0,0,-2,-2,-2,0,-1,1,-2,0,-2,
%T -4,0,4,-4,2,-2,2,-1,-4,4,2,1,2,2,0,6,0,0,4,-2,1,0,2,4,-8,-1,-2,2,-4,
%U 2,-2,1,-6,-4,-2,-1,-2,2,4,0,0,-2,-4,0,6,-6,0,-8,0,0,4
%N Expansion of eta(q^3) * eta(q^5) * eta(q^6) * eta(q^10) in powers of q.
%H Seiichi Manyama, <a href="/A030218/b030218.txt">Table of n, a(n) for n = 0..10000</a>
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%F Expansion of q * f(-q^3) * f(-q^5) * f(-q^6) * f(-q^10) in powers of q where f() is a Ramanujan theta function. - _Michael Somos_, Nov 17 2014
%F Euler transform of period 30 sequence [0, 0, -1, 0, -1, -2, 0, 0, -1, -2, 0, -2, 0, 0, -2, 0, 0, -2, 0, -2, -1, 0, 0, -2, -1, 0, -1, 0, 0, -4, ...]. - _Michael Somos_, Nov 17 2014
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 30 (t/i)^2 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Nov 17 2014
%F G.f. = x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(5*k)) * (1 - x^(6*k)) * (1 - x^(10*k)). - _Michael Somos_, Nov 17 2014
%F a(n) = -A286137(3*n). - _Michael Somos_, Mar 10 2020
%e G.f. = q - q^4 - q^6 - 2*q^7 + q^9 + q^10 - 2*q^11 + 2*q^12 + 2*q^14 - q^15 + ...
%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3] QPochhammer[ q^5] QPochhammer[ q^6] QPochhammer[ q^10], {q, 0, n}]; (* _Michael Somos_, Nov 17 2014 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^5 + A) * eta(x^6 + A) * eta(x^10 + A), n))}; /* _Michael Somos_, Nov 17 2014 */
%o (Magma) Basis( CuspForms( Gamma0(30), 2), 80) [1]; /* _Michael Somos_, Apr 27 2015 */
%Y Cf. A286137.
%K sign
%O 0,8
%A _N. J. A. Sloane_.