A187144 McKay-Thompson series of class 12I for the Monster group with a(0) = 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

G. C. Greubel, Table of n, a(n) for n = -1..1000

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

Cf. A058487, A123649.

Sequence in context: A284825 A318875 A187143 * A123635 A124304 A165408

Adjacent sequences: A187141 A187142 A187143 * A187145 A187146 A187147

Keyword

sign

Author

Michael Somos, Mar 05 2011

Status

approved