A030199 Expansion of x * Product_{k>=1} (1 - x^k) * (1 - x^(23*k)).

1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, -1, 1, 1, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1, -1, 1, 1, -1, 0, 0, 0, -1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, -1, 0, 0, -1

Offset

1,59

Comments

Number 40 of the 74 eta-quotients listed in Table I of Martin (1996).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

H. Darmon, Andrew Wiles's Marvelous Proof, Notices Amer. Math. Soc., 64 (No. 3, March 2017), 209-216. See p. 211 equ. 23

A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.

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.

Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3169-3176. MR1401749 (97m:11057)

J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 434.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Formula

Expansion of eta(q) * eta(q^23) in powers of q.

Euler transform of period 23 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, ...]. - Michael Somos, May 02 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 2 *u * v * w + 2 * u * w^2 + u^2 * w. - Michael Somos, May 02 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 * u3 * u6 +2 * u1 * u2 * u3 * u6 - 2 * u1 * u6^3 + 2 * u2^2 * u3 * u6 - u2 * u3^3. - Michael Somos, May 02 2005

a(n) is multiplicative with a(23^e) = 1. Let y = number of zeros of x^3 - x - 1 modulo p, then a(p^e) = (1 + (-1)^e)/2 if y = 1, a(p^e) = e+1 if y = 3, a(p^e) = (e-1)%3 - 1 if y = 0. - Michael Somos, Oct 19 2005

a(8*n + 4) = a(23*n + 5) = a(23*n + 7) = a(23*n + 10) = a(23*n + 11) = a(23*n + 14) = a(23*n + 15) = a(23*n + 19) = a(23*n + 20) = a(23*n + 21) = a(23*n + 22) = 0. - Michael Somos, Oct 19 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (23 t)) = 23^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 08 2014

2 * a(n) = A028959(n) - A028930(n). - Michael Somos, Sep 08 2014

Example

G.f. = q - q^2 - q^3 + q^6 + q^8 -q ^13 - q^16 + q^23 - q^24 + q^25 + q^26 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^23], {q, 0, n}]; (* Michael Somos, May 17 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^23 + A), n))}; /* Michael Somos, May 02 2005 */
(PARI) {a(n) = my(A, p, e, y); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==23, 1, y = sum( x=1, p-1, (x^3 - x - 1)%p == 0); if( y==1, 1-e%2, y, e+1, (e-1)%3 - 1))))}; /* Michael Somos, Oct 19 2005 */
(PARI) {a(n) = if( n<1, 0, qfrep([2, 1; 1, 12], n, 1)[n] - qfrep([4, 1; 1, 6], n, 1)[n])}; /* Michael Somos, Sep 08 2014 */
(Magma) Basis( CuspForms( Gamma1(23), 1), 82) [1]; /* Michael Somos, Sep 08 2014 */

Crossrefs

Cf. A028930, A028959.

Sequence in context: A165766 A102082 A358434 * A320005 A325414 A216510

Adjacent sequences: A030196 A030197 A030198 * A030200 A030201 A030202

Keyword

sign,mult

Author

N. J. A. Sloane

Extensions

Reference to Martin and Ono added by Chandan Singh Dalawat (dalawat(AT)gmail.com), Jul 23 2010

Status

approved