A169717 1A coefficients in an expansion of the elliptic genus of the K3 surface.

-1, 45, 231, 770, 2277, 5796, 13915, 30843, 65550, 132825, 260568, 494385, 915124, 1651815, 2922381, 5069867, 8650530, 14525742, 24053215, 39299778, 63447087, 101268540, 159963804, 250188435, 387746282, 595726956, 907877355, 1372935090, 2061208710, 3073155810, 4552039296, 6700526910

Offset: 0

Author

N. J. A. Sloane, Apr 19 2010

Keywords

• sign

Comments

Related to the Mathieu group M_24, see references.

Coefficients of the mock modular form H_1^(2). - N. J. A. Sloane, Mar 21 2015

Examples

			G.f. = -1 + 45*x + 231*x^2 + 770*x^3 + 2277*x^4 + 5796*x^5 + 13915*x^6 + ...
G.f. = -1/q + 45*q^7 + 231*q^15 + 770*q^23 + 2277*q^31 + 5796*q^39 + ...
		

References

• Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A Harvey, Umbral moonshine and the Niemeier lattices, Research in the Mathematical Sciences, 2014, 1:3; http://www.resmathsci.com/content/1/1/3
• Eguchi, T., Ooguri, H., Taormina, A., Yang, S. K., Superconformal algebras and string compactification on manifolds with SU(N) holonomy. Nucl. Phys. B315, 193 (1989). doi:10.1016/0550-3213(89)90454-9
• Eguchi, T., Taormina, A., Unitary representations of the N=4 superconformal algebra. Phys. Lett. B. 196(1), 75-81 (1987). doi:10.1016/0370-2693(87)91679-0
• Eguchi, T., Taormina, A., Character formulas for the N=4 superconformal algebra. Phys. Lett. B. 200(3), 315-322 (1988). doi:10.1016/0370-2693(88)90778-2
• H. Ooguri, Superconformal Symmetry and Geometry of Ricci Flat Kahler Manifolds, Int. J. Mod. Phys. A4 4303, 1989.

Links

• Miranda C. N. Cheng and John F. R. Duncan, On Rademacher sums, the largest Mathieu group, and the holographic modularity of moonshine (2011)
• Miranda C. N. Cheng and John F. R. Duncan, The largest Mathieu group and (mock) automorphic forms (2012)
• Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey, Umbral Moonshine, arXiv:1204.2779v3.pdf, Oct 13 2013.
• T. Eguchi and K. Hikami, Superconformal algebras and mock theta functions 2. Rademacher expansion for K3 surface, Commun. Number Theor. and Phys. 3, 531-554, 2009. [arXiv:0904.0911].
• Tohru Eguchi, Hirosi Ooguri and Yuji Tachikawa, Notes on the K3 surface and the Mathieu group M_24 (2010), arXiv:1004.0956; Exper. Math. 20, 91-96 (2011).

Crossrefs

Equals A212301/2.

Formula

a(n) ~ 2/sqrt(8*n - 1) * exp(2*Pi*sqrt(1/2*(n - 1/8))). This formula gives a good estimate of a(n) even at smaller values of n. [From N-E. Fahssi, Apr 26 2010]

Extensions

Added a(0)=-1 and further terms from Cheng et al. Umbral Moonshine paper. - N. J. A. Sloane, Mar 21 2015