A101127 McKay-Thompson series of class 12D for the Monster group.
1, 8, 28, 64, 134, 288, 568, 1024, 1809, 3152, 5316, 8704, 13990, 22208, 34696, 53248, 80724, 121240, 180068, 264448, 384940, 556064, 796760, 1132544, 1598789, 2243056, 3127360, 4333568, 5971922, 8188096, 11170160, 15163392, 20491033
Offset: 0
Keywords
Examples
T12D = 1/q + 8*q^2 + 28*q^5 + 64*q^8 + 134*q^11 + 288*q^14 + 568*q^17 + ...
References
- D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
- Index entries for McKay-Thompson series for Monster simple group
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
- Eric Weisstein's World of Mathematics, Infinite Product
Programs
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Mathematica
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^8, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *) a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^8, {x, 0, n}]; (* Michael Somos, Sep 12 2017 *) -
PARI
{a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^8, n))}; -
PARI
{a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, (n+1)\2, 1 + x^(2*k-1), 1 + A)^8, n))};
Formula
Expansion of q^(1/3) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^8 in powers of q.
Euler transform of period 4 sequence [8, -8, 8, 0, ...].
Given g.f. A(x), B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v*(u^3+v^3) -(u*v)^3 + 15*(u*v)^2 - 32*u*v + 16.
G.f.: (Product_{k>0} (1 + x^(2*k-1)))^8.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
Expansion of chi(x)^8 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 12 2017
G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018
Comments