A030200 Expansion of q^(-1/2) * eta(q) * eta(q^11) in powers of q.

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, 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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Comments

Number 52 of the 74 eta-quotients listed in Table I of Martin (1996).

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

Examples

			G.f. = 1 - x - x^2 + x^5 + x^7 - x^11 + x^13 - x^15 - x^16 - x^18 + 2*x^23 + ...
G.f. = q - q^3 - q^5 + q^11 + q^15 - q^23 + q^27 - q^31 - q^33 - q^37 + 2*q^47 +...
		

References

• F. Klein and R. Fricke, Vorlesungen ueber die theorie der elliptischen modulfunctionen, Teubner, Leipzig, 1892, Vol. 2, see p. 586.
• H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 203. MR1471703 (98g:14032)

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
• Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
• Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Crossrefs

Cf. A006571, A106276.

Programs

Magma

Basis( CuspForms( Gamma1(44), 1), 162) [1]; /* Michael Somos, Nov 13 2014 */

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^11], {x, 0, n}]; (* Michael Somos, Nov 12 2014 *)

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 */

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 */

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 */

Formula

Euler transform of period 11 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, ...]. - Michael Somos, Nov 20 2006

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

G.f.: Product_{k>0} (1 - x^k) * (1 - x^(11*k)).

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

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).

Convolution square is A006571.