A052241 McKay-Thompson series of class 8C for Monster.

1, 26, 79, 326, 755, 2106, 4460, 10284, 20165, 41640, 77352, 147902, 263019, 475516, 816065, 1413142, 2353446, 3936754, 6391091, 10390150, 16497734, 26184098, 40775677, 63394792, 97037170, 148178934, 223351867, 335704742, 499050461, 739575640, 1085723797

Offset

0,2

Comments

Let q = exp(-Pi*sqrt(58)/4). Then 396 = B(q) = 1/q + 26*q^3 + ... + a(n)*q^(4*n-1) + ... - Michael Somos, Sep 30 2019

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

Michael Somos, Emails to N. J. A. Sloane, 1993

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of 2 * q^(1/4) * ((k'^4 + 4*k^2) / (k'^2 * k))^(1/2) in powers of q. - Michael Somos, Sep 01 2014

Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - (u*v - 12) * (u*v - 32)^2. - Michael Somos, Sep 01 2014

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 01 2014

Convolution square is A007247. Convolution fourth power is A007267.

a(n) ~ exp(Pi*sqrt(2*n)) / (2^(5/4)*n^(3/4)). - Vaclav Kotesovec, May 01 2017

Example

G.f. = 1 + 26*x + 79*x^2 + 326*x^3 + 755*x^4 + 2106*x^5 + 4460*x^6 + ...
T8C = 1/q + 26*q^3 + 79*q^7 + 326*q^11 + 755*q^15 + 2106*q^19 + 4460*q^23 + ...

Mathematica

QP = QPochhammer; A = O[q]^40; A = (QP[q + A]/QP[q^2 + A])^12; s = Sqrt[A + 64*(q/A)]; CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, adapted from PARI *)
eta[q_] := q^(1/24)*QPochhammer[q]; e4D := q^(1/2)*(eta[q]/eta[q^2])^12;
T4B := e4D + 64*q/e4D; a[n_]:= SeriesCoefficient[Sqrt[(T4B /. {q -> q^2}) + O[q]^100], {q, 0, n}]; Table[a[n], {n, 0, 50}][[1 ;; ;; 2]] (* G. C. Greubel, Feb 13 2018 *)
a[ n_] := Module[ {m = InverseEllipticNomeQ @ q, A}, A = (1 - m) / (m / 16)^(1/2); SeriesCoefficient[ (A + 64/A)^(1/2), {q, 0, n - 1/4}]]; (* Michael Somos, Sep 30 2019 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( sqrt(A + 64 * x / A), n))}; /* Michael Somos, Sep 01 2014 */

Crossrefs

Cf. A007247, A007249, A007267.

Sequence in context: A042326 A042328 A042330 * A042332 A260199 A239172

Adjacent sequences: A052238 A052239 A052240 * A052242 A052243 A052244

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Status

approved