A030184 Expansion of eta(q) * eta(q^3) * eta(q^5) * eta(q^15) in powers of q.
1, -1, -1, -1, 1, 1, 0, 3, 1, -1, -4, 1, -2, 0, -1, -1, 2, -1, 4, -1, 0, 4, 0, -3, 1, 2, -1, 0, -2, 1, 0, -5, 4, -2, 0, -1, -10, -4, 2, 3, 10, 0, 4, 4, 1, 0, 8, 1, -7, -1, -2, 2, -10, 1, -4, 0, -4, 2, -4, 1, -2, 0, 0, 7, -2, -4, 12, -2, 0, 0, -8, 3, 10, 10, -1
Offset: 1
Examples
G.f. = q - q^2 - q^3 - q^4 + q^5 + q^6 + 3*q^8 + q^9 - q^10 - 4*q^11 + q^12 - 2*q^13 - ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
- N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 70.
- LMFDB, Space of modular forms of level 15 and weight 2.
- Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
- M. D. Rogers, Hypergeometric formulas for lattice sums and Mahler measures.
- Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Programs
-
Magma
Basis( CuspForms( Gamma1(15), 2), 76)[1]; /* Michael Somos, Nov 20 2014 */
-
Mathematica
a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^3] QPochhammer[ q^5] QPochhammer[ q^15], {q, 0, n}]; (* Michael Somos, Aug 11 2011 *) -
PARI
{a(n) = if( n<1, 0, ellak( ellinit([ 1, 1, 1, 0, 0], 1), n))}; /* Michael Somos, Aug 13 2006 */ -
PARI
{a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, (-1)^e, p==5, 1, a0=1; a1 = y = -if( p==2, 1, sum(x=0, p-1, kronecker( 4*x^3 + 5*x^2 + 2*x + 1, p))); for(i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 13 2006 */ -
PARI
{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^15 + A), n))}; /* Michael Somos, May 02 2005 */ -
Sage
CuspForms( Gamma1(15), 2, prec = 76).0; # Michael Somos, Aug 11 2011
Formula
Euler transform of period 15 sequence [ -1, -1, -2, -1, -2, -2, -1, -1, -2, -2, -1, -2, -1, -1, -4, ...]. - Michael Somos, May 02 2005
a(n) is multiplicative with a(5^e) = 1, a(3^e) = (-1)^e, otherwise a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) where a(p) = p minus number of points of elliptic curve modulo p. - Michael Somos, Aug 13 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 - u*w* (u + 2*v + 4*w). - Michael Somos, May 02 2005
G.f. A(x) satisfies 2 * A(x^2) = -(A(x) + A(-x) + 4*A(x^4)), A(x^2)^3 = -A(x) * A(-x) * A(x^4). - Michael Somos, Feb 19 2007
G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 - x^(5*k)) * (1 - x^(15*k)).
Comments