A003784 Coefficients of Jacobi cusp form of index 1 and weight 10.

0, 0, 0, 1, -2, 0, 0, -16, 36, 0, 0, 99, -272, 0, 0, -240, 1056, 0, 0, -253, -1800, 0, 0, 2736, -1464, 0, 0, -4284, 12544, 0, 0, -6816, -19008, 0, 0, 27270, -4554, 0, 0, -6864, 39880, 0, 0, -66013, -26928, 0, 0, 44064, 12544, 0, 0

Offset

0,5

References

M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, 1985, p. 141.

Links

Table of n, a(n) for n=0..50.

Formula

Expansion of eta(4z)^18 * theta_4(z) or (theta_2(z)^12 * theta_3(z)^3 * theta_4(z)^4) / 4096. - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 11 2000

Euler transform of period 4 sequence [ -2, -1, -2, -19, ...]. - Michael Somos, Mar 20 2004

Expansion of eta(q)^2 * eta(q^4)^18 / eta(q^2) in powers of q. - Michael Somos, Mar 20 2004

G.f.: x^3 * (Product_{k>0} (1 - x^k) * (1 - x^(4*k))^18 / (1 + x^k)). - Michael Somos, Mar 20 2004

a(4*n + 1) = a(4*n + 2) = 0.

G.f. for a(4*n + 3) = eta(q)^16 * eta(q^2)^5 / eta(q^4)^2; for a(4*n + 4) = -2 * eta(q)^18 * eta(q^4)^2 / eta(q^2). - Michael Somos, Mar 20 2004

Example

q^3 - 2*q^4 - 16*q^7 + 36*q^8 + 99*q^11 - 272*q^12 - 240*q^15 + 1056*q^16 + ...

Mathematica

QP = QPochhammer; s = q^3*QP[q]^2*(QP[q^4]^18/QP[q^2]) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 29 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<3, 0, n-=3; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^18 / eta(x^2 + A), n))} /* Michael Somos, Mar 20 2004 */

Crossrefs

Cf. A003785.

Sequence in context: A003193 A108474 A120582 * A368849 A244143 A066294

Adjacent sequences: A003781 A003782 A003783 * A003785 A003786 A003787

Keyword

sign

Author

N. J. A. Sloane

Status

approved