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A112146
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McKay-Thompson series of class 9b for the Monster group.
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6
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1, 0, 9, -4, 0, 36, 2, 0, 126, 12, 0, 324, -21, 0, 801, 4, 0, 1764, 36, 0, 3744, -68, 0, 7452, 21, 0, 14400, 112, 0, 26748, -184, 0, 48510, 44, 0, 85536, 275, 0, 147924, -456, 0, 250452, 112, 0, 417276, 644, 0, 683640, -1019, 0, 1104948, 240, 0, 1761552, 1370, 0
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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 q^(1/3) * 3*( b(q) / c(q) + c(q) / b(q)) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Mar 24 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u+v)^3 - (u^2 + 3*u - 18) * (v^2 + 3*v - 18).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + w^2 + u*w + 18*(u+w) - (w+u)*v^2 - 9*v + 54.
Expansion of ( (eta(q^3) / eta(q^9))^4 + 9 * (eta(q^9) / eta(q^3))^4) in powers of q.
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EXAMPLE
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T9b = 1/q + 9*q - 4*q^2 + 36*q^4 + 2*q^5 + 126*q^7 + 12*q^8 + 324*q^10 + ...
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MATHEMATICA
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QP = QPochhammer; s = (QP[q^3]^8 + 9*q^2*QP[q^9]^8)/(QP[q^3]^4*QP[q^9]^4) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
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PROG
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(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^3 + A) / eta(x^9 + A))^4; polcoeff( A + 9*x^2 / A, n))}; /* Michael Somos, Mar 24 2007 */
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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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