cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A003785 Coefficients of Jacobi cusp form of index 1 and weight 12.

Original entry on oeis.org

1, 10, 0, 0, -88, -132, 0, 0, 1275, 736, 0, 0, -8040, -2880, 0, 0, 24035, 13080, 0, 0, -14136, -54120, 0, 0, -128844, 115456, 0, 0, 389520, 38016, 0, 0, -256410, -697950, 0, 0, -806520, 963160, 0, 0, 1892363, 938400, 0, 0, -1227600, -2309120, 0, 0, -813450, -2813096, 0, 0
Offset: 3

Views

Author

N. J. A. Sloane, Mar 15 1996

Keywords

Examples

			q^3 + 10*q^4 - 88*q^7 - 132*q^8 + 1275*q^11 + 736*q^12 - 8040*q^15 - ...
		

References

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

Crossrefs

Cf. A003784.

Programs

  • PARI
    {a(n) = local(A, A1); if( n<3, 0, n -= 3; A = x * O(x^n); A1 = (eta(x^2 + A)^3 / eta(x + A) / eta(x^4 + A)^2)^4 ; polcoeff( (A1 + 4 * x / A1) * eta(x^2 + A)^7 * eta(x^4 + A)^18 / eta(x + A)^2, n))} /* Michael Somos, Oct 24 2007 */

Formula

(theta_3(z)^4+(theta_2(z)^4)/4)*eta(4z)^18*theta_4(z). - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 11 2000
a(4*n+1) = a(4*n+2) = 0.

A055978 A sequence related to Ramanujan's tau function.

Original entry on oeis.org

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

Views

Author

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

Keywords

Examples

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

References

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

Crossrefs

Programs

  • 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 */

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.
Showing 1-2 of 2 results.