A212301 -id:A212301 - cp's OEIS frontend

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

N. J. A. Sloane, Apr 19 2010

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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

			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 + ...

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.

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).

Equals A212301/2.

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]

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

A212312 4 times coefficients of the twisted K3 elliptic genus attached to the conjugacy class 7A or 7B (7AB) in the Mathieu group M_24.

Original entry on oeis.org

-1, 213, 4767, 52682, 627573, 4682412, 53313931, 341278995, 3421674678, 24687536097, 214728131688, 1291490582217, 14602976386348, 80718169058895, 691486905294549, 5191979939244635, 44020035323951778, 245857487857551822, 2489942402834854279, 13633673140696254618

Offset: 1

Charles R Greathouse IV and John F. R. Duncan, Aug 02 2012

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Miranda C. N. Cheng and John F. R. Duncan, On Rademacher sums, the largest Mathieu group, and the holographic modularity of moonshine, arXiv:1110.3859 [math.RT], 2011.
Miranda C. N. Cheng and John F. R. Duncan, The largest Mathieu group and (mock) automorphic forms, arXiv:1201.4140 [math.RT], 2012.
Tohru Eguchi and Kazuhiro Hikami, Note on Twisted Elliptic Genus of K3 Surface, arXiv:1008.4924 [hep-th], 2010.

Other conjugacy classes: A212301-A212320.

F22(q, prec)={
    sum(s=1,min(prec-1,sqrt(2*prec)-1/2),
        my(t=0);
        forstep(r=s+1,2*prec\s,2,
            t+=(-1)^r*q^(r*s/2)
        );
        s*t
    )
};
list(n)={
    my(q='q+O('q^(n+1)),e=eta(q),f=F22(q,n),ser);
    ser=(6*sum(k=1, n, k*q^k/(1-q^k))+6*f-7*q*log(eta(7*q)/e)'-1/4)/e^3;
    Vec(4*ser)
};

cp's OEIS Frontend

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

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