A030207 Expansion of eta(q)^2 * eta(q^2) * eta(q^4) * eta(q^8)^2 in powers of q.
1, -2, -2, 4, 0, 4, 0, -8, -5, 0, 14, -8, 0, 0, 0, 16, 2, 10, -34, 0, 0, -28, 0, 16, 25, 0, 28, 0, 0, 0, 0, -32, -28, -4, 0, -20, 0, 68, 0, 0, -46, 0, 14, 56, 0, 0, 0, -32, 49, -50, -4, 0, 0, -56, 0, 0, 68, 0, -82, 0, 0, 0, 0, 64, 0, 56, 62, 8, 0, 0, 0, 40, -142, 0, -50, -136, 0, 0, 0, 0, -11, 92, 158, 0, 0, -28, 0
Offset: 1
Examples
G.f. = q - 2*q^2 - 2*q^3 + 4*q^4 + 4*q^6 - 8*q^8 - 5*q^9 + 14*q^11 - 8*q^12 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
- M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
- M. Koike, Mathieu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
- 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
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
-
Magma
Basis( CuspForms( Gamma1(8), 3), 100) [1]; /* Michael Somos, May 27 2014 */
-
Mathematica
a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4] QPochhammer[ q^8]^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *) a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 3, 0, q^2] EllipticTheta[ 2, 0, q^2]^2, {q, 0, n}]; (* Michael Somos, May 17 2015 *) a[ n_] := SeriesCoefficient[ (-EllipticTheta[ 3, 0, q]^5 EllipticTheta[ 3, 0, q^2] + 3 EllipticTheta[ 3, 0, q]^3 EllipticTheta[ 3, 0, q^2]^3 - 2 EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2]^5) / 4, {q, 0, n}]; (* Michael Somos, May 17 2015 *) -
PARI
{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^8 + A))^2 * eta(x^2 + A) * eta(x^4 + A), n))}; /* Michael Somos, May 28 2007 */ -
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==2, (-2)^e, p%8>4, if( e%2, 0, p^e), for( x=1, sqrtint(p\2), if( issquare( p - 2*x^2, &y), break)); y = 4*y^2 - 2*p; a0=1; a1=y; for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Jun 13 2007 */ -
Sage
CuspForms( Gamma1(8), 3, prec = 100).0; # Michael Somos, Aug 11 2011
Formula
Expansion of q * phi(q) * phi(-q)^2 * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, May 28 2007
Expansion of (3 * phi(q)^3 * phi(q^2)^3 - 2 * phi(q) * phi(q^2)^5 - phi(q)^5 * phi(q^2)) / 4 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jun 13 2007
Euler transform of period 8 sequence [-2, -3, -2, -4, -2, -3, -2, -6, ...]. - Michael Somos, May 28 2007
a(n) is multiplicative with a(2^e) = (-2)^e, a(p^e) = (1+(-1)^e)/2 * p^e if p == 5, 7 (mod 8), a(p^e) = a(p)*a(p^(e-1)) - p^2*a(p^(e-2)) if p == 1, 3 (mod 8) where a(p) = 4*x^2 -2*p and p = x^2 + 2*y^2. - Michael Somos, Jun 13 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 512^(1/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 25 2007
G.f.: (1/2) * Sum_{u,v in Z} (u*u - 2*v*v) * x^(u*u + 2*v*v). - Michael Somos, Jun 14 2007
G.f.: x * Product_{k>0} (1 - x^k)^6 * (1 + x^k)^4 * (1 + x^(2*k))^3 * (1 + x^(4*k))^6. - Michael Somos, May 28 2007
Comments