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A187147 McKay-Thompson series of class 12B for the Monster group with a(0) = -4. 1
1, -4, 6, -4, -3, 12, -8, -12, 30, -20, -30, 72, -46, -60, 156, -96, -117, 300, -188, -228, 552, -344, -420, 1008, -603, -732, 1770, -1048, -1245, 2976, -1776, -2088, 4908, -2900, -3420, 7992, -4658, -5460, 12756, -7408, -8583, 19944, -11564, -13344, 30756 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,2

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0. .oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

REFERENCES

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

LINKS

Table of n, a(n) for n=-1..43.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (1/q) * (psi(-q) / psi(-q^3))^4 in powers of q.

Expansion of (eta(q) * eta(q^4) * eta(q^6) / (eta(q^2) * eta(q^3) * eta(q^12)))^4 in powers of q.

Euler transform of period 12 sequence [ -4, 0, 0, -4, -4, 0, -4, -4, 0, 0, -4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 9 / f(t) where q = exp(2 pi i t).

Convolution square of A062243.

G.f.: ( Product_{k>0} (1 - x^(4*k)) * (1 - x^(2*k-1)) / (1 - x^(3*k)) )^4.

EXAMPLE

1/q - 4 + 6*q - 4*q^2 - 3*q^3 + 12*q^4 - 8*q^5 - 12*q^6 + 30*q^7 + ...

PROG

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^4, n))}

CROSSREFS

Cf. A062243, A112148.

Sequence in context: A204693 A204817 A199721 * A128633 A001482 A198493

Adjacent sequences:  A187144 A187145 A187146 * A187148 A187149 A187150

KEYWORD

sign

AUTHOR

Michael Somos, Mar 05 2011

STATUS

approved

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Last modified November 27 17:39 EST 2014. Contains 250249 sequences.