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A187143
McKay-Thompson series of class 12I for the Monster group with a(0) = -1.
2
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
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (1/q) * (psi(q^2) * psi(q^3)^3) / (psi(q) * psi(q^6)^3) in powers of q where psi() is a Ramanujan theta function.
Expansion of eta(q) * eta(q^4)^2 * eta(q^6)^9 / (eta(q^2)^3 * eta(q^3)^3 * eta(q^12)^6) in powers of q.
Euler transform of period 12 sequence [ -1, 2, 2, 0, -1, -4, -1, 0, 2, 2, -1, 0, ...].
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]*QP[q^4]^2*(QP[q^6]^9/(QP[q^2]^3*QP[q^3]^3* QP[q^12]^6)) + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^9 / (eta(x^2 + A)^3 * eta(x^3 + A)^3 * eta(x^12 + A)^6), n))};
CROSSREFS
Cf. A058487.
Sequence in context: A110270 A284825 A318875 * A187144 A123635 A376505
KEYWORD
sign
AUTHOR
Michael Somos, Mar 05 2011
STATUS
approved