A055978 A sequence related to Ramanujan's tau function.

1, -2, 0, 4, -24, 36, 0, -64, 252, -290, 0, 396, -1472, 1380, 0, -944, 4830, -4248, 0, -1268, -6048, 8040, 0, 12528, -16744, -3706, 0, -20976, 84480, -31284, 0, -31312, -113643, 101542, 0, 152892, -115920, -104792, 0, -96576, 534612, -112914, 0, -369544, -370944, 334864, 0, 603936, -577738, -22554, 0

Offset

4,2

References

Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser, 1985.

Links

Table of n, a(n) for n=4..54.

Formula

a(4*n + 2) = 0, a(4*n) = A000594(n) (Ramanujan tau(n)).

Sum_{k>0} a(4*k + 1) * q^(4*k + 1) = (-1) * (q * d/dq theta_2(q^4)) * eta(q^4)^18 * eta(q^16)^2 / eta(q^8). - Michael Somos, Mar 20 2004

Sum_{k>0} a(4*k + 3) * q^(4*k + 3) = (1/2) * (q * d/dq theta_3(q^4)) * eta(q^4)^16 * eta(q^8)^5 / eta(q^16)^2. - Michael Somos, Mar 20 2004

G.f.: x^3 * (Product_{k>0} (1 - x^k) * (1 - x^(4*k))^18 / (1 + x^k)) * (Sum_{k>0} k^2 * x^(k^2)). - Michael Somos, Mar 20 2004

phi_{10, 1}*q*(d/dq){theta_3(z)} where phi_{10, 1} is unique Jacobi cusp form of weight 10 index 1 given by A003784.

Example

q^4 - 2*q^5 + 4*q^7 - 24*q^8 + 36*q^9 - 64*q^11 + 252*q^12 - 290*q^13 + ...

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) * sum( k=1, sqrtint(n), k^2 * x^(k^2)), n))} /* Michael Somos, Mar 20 2004 */

Crossrefs

A003784, A000594.

Sequence in context: A326215 A248643 A199852 * A356189 A245695 A374286

Adjacent sequences: A055975 A055976 A055977 * A055979 A055980 A055981

Keyword

sign

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 24 2000

Status

approved