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A058095
McKay-Thompson series of class 9c for the Monster group.
4
1, -4, 2, 12, -21, 4, 36, -68, 21, 112, -184, 44, 275, -456, 112, 644, -1019, 240, 1370, -2156, 514, 2828, -4340, 992, 5498, -8392, 1930, 10428, -15675, 3528, 19060, -28472, 6399, 34072, -50382, 11184, 59333, -87260, 19312, 101496, -148148, 32480, 170130, -247156
OFFSET
0,2
COMMENTS
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
FORMULA
Expansion of q^(1/3) * 3 * b(q) / c(q) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of q^(1/3) * (eta(q) / eta(q^3))^4 in powers of q. - Michael Somos, Mar 24 2007
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+v)^3 - u*v * (u+3) * (v+3) .
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*v^2 + v^2*w^2 - v*u^2*w^2 + u*w*v^2 - 9*u*w * (u+w).
G.f.: (Product_{k>0} (1 + x^k + x^(2*k)))^-4.
Euler transform of period 3 sequence [ -4, -4, 0, ...]. - Michael Somos, Mar 24 2007
a(n) = A112146(3*n - 1). Convolution inverse of A128758.
EXAMPLE
G.f. = 1 - 4*x + 2*x^2 + 12*x^3 - 21*x^4 + 4*x^5 + 36*x^6 - 68*x^7 + ...
T9c = 1/q - 4*q^2 + 2*q^5 + 12*q^8 - 21*q^11 + 4*q^14 + 36*q^17 - 68*q^20 + ...
MATHEMATICA
a[0] = 1; a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[(QPochhammer[x + A]/QPochhammer[x^3 + A])^4, n]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 06 2015, adapted from Michael Somos's PARI script *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^3 + A))^4, n))}; /* Michael Somos, Mar 24 2007 */
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Nov 27 2000
STATUS
approved