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

0,2

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

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

Table of n, a(n) for n=0..31.

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).

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]

Example

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 + ...

Crossrefs

Equals A212301/2.

Sequence in context: A280059 A251451 A251444 * A246420 A172118 A127073

Adjacent sequences: A169714 A169715 A169716 * A169718 A169719 A169720

Keyword

sign

Author

N. J. A. Sloane, Apr 19 2010

Extensions

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

Status

approved