|
%I #12 Jan 10 2023 19:27:18
%S 1,-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,
%T 0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,-1,1,1,0,1,-1,0,-1,0,0,1,0,1,0,0,1,0,
%U 0,0,0,0,0,-1,-1,0,0,-1,0,0,0,-1,0,0,0,0
%N Expansion of eta(q) * eta(q^47) in powers of q.
%H G. C. Greubel, <a href="/A256538/b256538.txt">Table of n, a(n) for n = 2..2500</a>
%H Akihiko Okamoto, <a href="http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1617-14.pdf">On expression of eta-product by theta series</a>, RIMS Kokyuroku (1617), 157-166, 2008-10.
%F Euler transform of a period 47 sequence.
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^6 +2*w*v^4*u +4*w^2*v^3*u +2*w*v^3*u^2 +8*w^2*v^2*u^2 +2*w^2*v*u^3 +4*w^3*v*u^2 -4*w^4*u^2 +4*w^3*u^3 -w^2*u^4.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (47 t)) = 47^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F G.f.: x^2 * Product{k>0} (1 - x^k) * (1 - x^(47*k)).
%e G.f. = q^2 - q^3 - q^4 + q^7 + q^9 - q^14 - q^17 + q^24 + q^28 - q^37 - q^42 + ...
%t a[ n_] := SeriesCoefficient[ q^2 QPochhammer[ q] QPochhammer[ q^47], {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^47 + A), n))};
%o (PARI) {a(n) = if( n<1, 0, qfrep([4, 1; 1, 12], n, 1)[n] - qfrep([6, 1; 1, 8], n, 1)[n])};
%o (Magma) Basis( CuspForms( Gamma1(47), 1), 84) [2];
%Y Cf. A030199.
%K sign
%O 2,165
%A _Michael Somos_, Apr 01 2015
|
