|
%I #30 Sep 08 2022 08:44:50
%S 1,-1,-1,0,0,1,0,1,0,0,0,-1,0,1,0,-1,-1,0,-1,0,0,0,0,2,1,0,2,-1,0,-1,
%T 0,0,0,-1,1,-1,0,0,0,0,-1,0,0,0,-1,0,1,0,-1,0,0,2,0,0,0,1,-1,1,0,0,1,
%U 0,1,0,0,0,0,-1,-1,0,-2,0,0,-1,0,0,0,1,-1,-2,0,2,1,0,1,0,0,0,1,-1,-1,0,1,0,0,-1,0,0,0,2,1,0,0,0,0
%N Expansion of q^(-1/2) * eta(q) * eta(q^11) in powers of q.
%C Number 52 of the 74 eta-quotients listed in Table I of Martin (1996).
%C In [Klein and Fricke 1892] on page 586 equation (3) first line left side has A_0 and the right side the power series r^{1/2} (1 - r - r^2 + r^5 + r^7 + ...) which is the g.f. of this sequence. A_0 and the other A_1, A_3, A_9, A_5, A_4 (in a permuted order) correspond to the nonzero 11-sections of the g.f. of this sequence. - _Michael Somos_, Nov 12 2014
%D F. Klein and R. Fricke, Vorlesungen ueber die theorie der elliptischen modulfunctionen, Teubner, Leipzig, 1892, Vol. 2, see p. 586.
%D H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 203. MR1471703 (98g:14032)
%H Seiichi Manyama, <a href="/A030200/b030200.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.
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%F Euler transform of period 11 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, ...]. - _Michael Somos_, Nov 20 2006
%F a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(11^e) = 1, b(p^e) = (e-1)%3 - 1 if f=0, b(p^e) = e+1 if f=3, b(p^e) = (1 + (-1)^e) / 2 if f=1 where f = number of zeros of x^3 - x^2 - x - 1 modulo p. - _Michael Somos_, Nov 20 2006
%F G.f.: Product_{k>0} (1 - x^k) * (1 - x^(11*k)).
%F a(n) = sum over all solutions to x^2 + x*y + 3*y^2 = 2*n + 1 with odd integer x>0 of (-1)^y. - _Michael Somos_, Jan 29 2007
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F Convolution square is A006571.
%e G.f. = 1 - x - x^2 + x^5 + x^7 - x^11 + x^13 - x^15 - x^16 - x^18 + 2*x^23 + ...
%e G.f. = q - q^3 - q^5 + q^11 + q^15 - q^23 + q^27 - q^31 - q^33 - q^37 + 2*q^47 +...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^11], {x, 0, n}]; (* _Michael Somos_, Nov 12 2014 *)
%o (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; qfrep( [1, 0; 0, 11], n)[n] - qfrep( [3, 1; 1, 4], n)[n])}; /* _Michael Somos_, Nov 20 2006 */
%o (PARI) {a(n) = my(A, p, e, f); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==11, 1, f = sum( k=0, p-1, (k^3 - k^2 - k - 1)%p == 0); if( f==0, (e-1)%3-1, if( f==1, (1 + (-1)^e) / 2, e+1)))))}; /* _Michael Somos_, Nov 20 2006 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^11 + A), n))}; /* _Michael Somos_, Nov 20 2006 */
%o (Magma) Basis( CuspForms( Gamma1(44), 1), 162) [1]; /* _Michael Somos_, Nov 13 2014 */
%Y Cf. A006571, A106276.
%K sign
%O 0,24
%A _N. J. A. Sloane_
|
