A058091 McKay-Thompson series of class 9B for the Monster group.
1, 5, -7, 3, 15, -32, 9, 58, -96, 22, 149, -253, 68, 372, -599, 140, 826, -1317, 317, 1768, -2735, 632, 3526, -5434, 1259, 6854, -10364, 2346, 12765, -19188, 4345, 23224, -34524, 7693, 41049, -60654, 13487, 71176, -104303, 22962, 120718, -176050, 38622, 201539
Offset: 0
Examples
G.f. = 1 + 5*x - 7*x^2 + 3*x^3 + 15*x^4 - 32*x^5 + 9*x^6 + 58*x^7 - 96*x^8 + ... T9B = 1/q + 5*q^2 - 7*q^5 + 3*q^8 + 15*q^11 - 32*q^14 + 9*q^17 + 58*q^20 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
- J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
- Index entries for sequences related to groups
- Index entries for McKay-Thompson series for Monster simple group
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] / QPochhammer[ q^9])^3 + 3, {q, 0, 3 n - 1}]; (* Michael Somos, Aug 20 2014 *) -
PARI
{a(n) = local(A); if( n<0, 0, n*=3; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^9 + A))^3, n))}; /* Michael Somos, Aug 09 2006 */ -
PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = eta(x + A) / eta(x^2 + A); A = subst(A + x * O(x^(n\3)), x, x^3)^3 / A; polcoeff( A + 4*x / A^2, n))}; /* Michael Somos, Aug 09 2006 */
Formula
Expansion of 3 * q^(1/3) * a(q) / c(q) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Aug 09 2006
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^3 + v^3 - u^2*v^2 + 9*u*v - 54. - Michael Somos, Aug 09 2006
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^3 + 1) * (v^3 + 1) - (u*v + 5) * (u^2*v^2 - 4*u*v + 11). - Michael Somos, Aug 20 2014
G.f.: Sum_{k>=0} a(k) * x^(3*k) = 3*x + (Product_{k>0} (1 - x^k) / (1 - x^(9*k)))^3. - Michael Somos, Aug 09 2006
a(n) = A131986(3*n - 1). - Michael Somos, Aug 20 2014
Comments