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

Author

N. J. A. Sloane, Nov 27 2000

Keywords

• nonn

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

Examples

			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 + ...
		

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

Crossrefs

Cf. A007247, A007249, A007267.

Programs

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 *)

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 */

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