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A187144
McKay-Thompson series of class 12I for the Monster group with a(0) = 1.
1
1, 1, 2, 0, 1, 0, 0, 0, -2, 0, -2, 0, 2, 0, 4, 0, 3, 0, -4, 0, -8, 0, -4, 0, 5, 0, 14, 0, 7, 0, -8, 0, -20, 0, -12, 0, 14, 0, 28, 0, 17, 0, -20, 0, -44, 0, -24, 0, 28, 0, 66, 0, 36, 0, -40, 0, -90, 0, -52, 0, 56, 0, 124, 0, 71, 0, -80, 0, -176, 0, -96, 0, 109, 0, 244, 0, 133, 0, -144
OFFSET
-1,3
COMMENTS
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
FORMULA
Expansion of c(q) / c(q^4) in powers of q where c() is a cubic AGM function.
Expansion of eta(q^3)^3 * eta(q^4) / (eta(q) * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [ 1, 1, -2, 0, 1, -2, 1, 0, -2, 1, 1, 0, ...].
Convolution inverse of A123649.
a(2*n) = 0 unless n=0. a(2*n - 1) = A058487(n).
EXAMPLE
G.f. = 1/q + 1 + 2*q + q^3 - 2*q^7 - 2*q^9 + 2*q^11 + 4*q^13 + 3*q^15 - 4*q^17 + ...
MATHEMATICA
QP = QPochhammer; s = QP[q^3]^3*(QP[q^4]/(QP[q]*QP[q^12]^3)) + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A) / (eta(x + A) * eta(x^12 + A)^3), n))};
CROSSREFS
Sequence in context: A284825 A318875 A187143 * A123635 A376505 A124304
KEYWORD
sign
AUTHOR
Michael Somos, Mar 05 2011
STATUS
approved