A013953 Expansion of the modular form of level 4 and weight 1/2.

1, 0, 0, 4, -240, 0, 0, 26760, -85995, 0, 0, 1707264, -4096240, 0, 0, 44330496, -91951146, 0, 0, 708938760, -1343913984, 0, 0, 8277534720, -14733025125, 0, 0, 77092288000, -130880766192, 0, 0, 604139268096, -988226335125, 0, 0, 4125992712192, -6548115718144, 0, 0, 25168873498760

Offset

-3,4

Links

Table of n, a(n) for n=-3..36.

R. E. Borcherds, Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.

J. H. Bruinier, Infinite products in number theory and geometry, arXiv:math/0404427 [math.NT], 2004.

K. Ono, The last words of a genius, Notices Amer. Math. Soc., 57 (2010), 1410-1419.

Formula

60*theta_3(z)+KZ(z)*E_6(4z)/del_12(4z) where KZ(z) is the cusp form of weight 13/2 given by the sequence A054891. - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jan 11 2001

a(4*n + 2) = a(4*n + 3) = 0. - Michael Somos, Feb 22 2015

Example

G.f. = 1/q^3 + 4 - 240*q + 26760*q^4 - 85995*q^5 + 1707264*q^8 - 4096240*q^9 + ...

Prog

(PARI) {a(n) = my(A, F, t, T, E); if( n<-3, 0, n += 4; A = x * O(x^n); t = sum( k= 1, sqrtint( n), 2 * x^k^2, 1 + A); T = t^20; E = sum( k= 1, n\4, -264 * sigma( k, 9) * x^(4*k), 1 + A); polcoeff( (( E / T )' * T / eta(x^4 + A)^24 + 1056*x^3) * -1/40 * t, n-1))}; /* Michael Somos, Jul 08 2011 */

Crossrefs

Cf. A000025, A192732.

Sequence in context: A132551 A333864 A358158 * A051753 A323996 A094073

Adjacent sequences: A013950 A013951 A013952 * A013954 A013955 A013956

Keyword

sign

Author

N. J. A. Sloane.

Extensions

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jan 11 2001

Status

approved