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A112146 McKay-Thompson series of class 9b for the Monster group. 6
1, 0, 9, -4, 0, 36, 2, 0, 126, 12, 0, 324, -21, 0, 801, 4, 0, 1764, 36, 0, 3744, -68, 0, 7452, 21, 0, 14400, 112, 0, 26748, -184, 0, 48510, 44, 0, 85536, 275, 0, 147924, -456, 0, 250452, 112, 0, 417276, 644, 0, 683640, -1019, 0, 1104948, 240, 0, 1761552, 1370, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
-1,3
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) + c(q) / b(q)) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Mar 24 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u+v)^3 - (u^2 + 3*u - 18) * (v^2 + 3*v - 18).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + w^2 + u*w + 18*(u+w) - (w+u)*v^2 - 9*v + 54.
Expansion of ( (eta(q^3) / eta(q^9))^4 + 9 * (eta(q^9) / eta(q^3))^4) in powers of q.
a(3*n) = 0. a(3*n-1) = A058095(n). a(3*n + 1) = 9 * A128758(n). - Michael Somos, Feb 19 2015
EXAMPLE
T9b = 1/q + 9*q - 4*q^2 + 36*q^4 + 2*q^5 + 126*q^7 + 12*q^8 + 324*q^10 + ...
MATHEMATICA
QP = QPochhammer; s = (QP[q^3]^8 + 9*q^2*QP[q^9]^8)/(QP[q^3]^4*QP[q^9]^4) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^3 + A) / eta(x^9 + A))^4; polcoeff( A + 9*x^2 / A, n))}; /* Michael Somos, Mar 24 2007 */
CROSSREFS
Sequence in context: A307215 A354636 A021918 * A056897 A263192 A355282
KEYWORD
sign
AUTHOR
Michael Somos, Aug 28 2005, Aug 09 2008
STATUS
approved

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Last modified March 19 07:25 EDT 2024. Contains 370955 sequences. (Running on oeis4.)