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A045488
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McKay-Thompson series of class 6E for the Monster group with a(0) = 1.
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4
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1, 1, 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, 17744, -20448, -46944, -26916
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OFFSET
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-1,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of (1/q) * a(q^2) * psi(q) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function and a() is a cubic AGM theta function. - Michael Somos, May 22 2015
Expansion of 6 + eta(q)^5 * eta(q^3) / (eta(q^2) * eta(q^6)^5) in powers of q. - Michael Somos, May 22 2015
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EXAMPLE
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G.f. = 1/q + 1 + 6*q + 4*q^2 - 3*q^3 - 12*q^4 - 8*q^5 + 12*q^6 + 30*q^7 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 6 + QPochhammer[ q]^5 QPochhammer[ q^3] / (q QPochhammer[ q^2] QPochhammer[ q^6]^5), {q, 0, n}]; (* Michael Somos, May 22 2015 *)
a[ n_] := SeriesCoefficient[ -2 + (1/q) (QPochhammer[ q^2] QPochhammer[ q^3]^3 / (QPochhammer[ q] QPochhammer[ q^6]^3))^3, {q, 0, n}]; (* Michael Somos, May 22 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^3 / EllipticTheta[ 3, 0, q^3] + 3 EllipticTheta[ 3, 0, q^3]^3 / EllipticTheta[ 3, 0, q]) EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q^(3/2)]^3, {q, 0, n}]; (* Michael Somos, May 22 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 6*x + eta(x + A)^5 * eta(x^3 + A) / (eta(x^2 + A) * eta(x^6 + A)^5), n))}; /* Michael Somos, May 22 2015 */
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( -2*x + (eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3))^3, n))}; /* Michael Somos, May 22 2015 */
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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