A058511 McKay-Thompson series of class 15D for the Monster group.
1, -2, -1, 2, 1, 4, -6, -2, 2, 0, 10, -14, -5, 8, 4, 20, -28, -10, 14, 4, 39, -56, -20, 28, 10, 72, -100, -34, 46, 16, 128, -176, -61, 86, 30, 216, -294, -100, 134, 44, 355, -484, -165, 226, 79, 568, -770, -260, 350, 116, 894, -1208, -408, 552, 188, 1376, -1848, -620, 830, 276, 2087, -2800, -940
Offset: 0
Keywords
Examples
G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 4*x^5 - 6*x^6 - 2*x^7 + 2*x^8 + ... T15D = 1/q - 2*q^2 - q^5 + 2*q^8 + q^11 + 4*q^14 - 6*q^17 - 2*q^20 + 2*q^23 + ...
Links
- Vaclav Kotesovec, 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.
- Index entries for McKay-Thompson series for Monster simple group
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^5])^2, {x, 0, n}]; (* Michael Somos, Aug 26 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^5 + A))^2, n))}; /* Michael Somos, Dec 17 2010 */
Formula
Expansion of q^(1/3) * (eta(q) / eta(q^5))^2 in powers of q.
Euler transform of period 5 sequence [ -2, -2, -2, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (45 t)) = 5 / f(t) where q = exp(2 Pi i t).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (v - u^2) + 4*u*v.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + u*w + w^2 - v^2 * (u + w) - 5*v.