OFFSET
0,24
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
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.
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.
EXAMPLE
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 +...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^11], {x, 0, n}]; (* Michael Somos, Nov 12 2014 *)
PROG
(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 */
(Magma) Basis( CuspForms( Gamma1(44), 1), 162) [1]; /* Michael Somos, Nov 13 2014 */
CROSSREFS
KEYWORD
sign
AUTHOR
STATUS
approved