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A058483
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McKay-Thompson series of class 12E for the Monster group.
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3
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1, -1, 7, -9, 10, -23, 38, -47, 75, -112, 148, -217, 293, -385, 553, -728, 928, -1272, 1670, -2111, 2765, -3566, 4504, -5784, 7300, -9123, 11592, -14458, 17838, -22342, 27668, -33884, 41843, -51344, 62548, -76515, 92989, -112514, 136687, -164961, 198190, -238991
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history;
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OFFSET
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0,3
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COMMENTS
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Given g.f. A(x), B(q) = q*A(q^2) satisfies 0 = f(B(q). B(q^2)) where f(u, v) = 12 + v^2 - 2*u^2 - u^2*v. - Michael Somos, Apr 21 2004
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LINKS
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FORMULA
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a(n) ~ (-1)^n * exp(sqrt(2*n/3)*Pi) / (2^(5/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017
Expansion of F - 3*q/F, where F = q^(1/2)*(eta(q^2)^2 * eta(q^3)/(eta(q) * eta(q^6)^2))^2 in powers of q. - G. C. Greubel, Jun 03 2018
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EXAMPLE
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G.f. = 1 - x + 7*x^2 - 9*x^3 + 10*x^4 - 23*x^5 + 38*x^6 - 47*x^7 + ...
T12E = 1/q - q + 7*q^3 - 9*q^5 + 10*q^7 - 23*q^9 + 38*q^11 - 47*q^13 + ...
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MATHEMATICA
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QP = QPochhammer; A = O[q]^50; A = ((QP[q^2 + A]^2*QP[q^3 + A])/(QP[q + A]* QP[q^6 + A]^2))^2; s = A - 3*(q/A); CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, adapted from PARI *)
eta[q_]:= q^(1/24)*QPochhammer[q]; F:= q^(1/2)*(eta[q^2]^2*eta[q^3]/(eta[q]*eta[q^6]^2))^2; a := CoefficientList[Series[F - 3*q^1/F, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 03 2018 *)
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x^2 * O(x^n); A = ((eta(x^2 + A)^2 * eta(x^3 + A)) / (eta(x + A) * eta(x^6 + A)^2))^2; polcoeff( A - 3*x / A, n))}; /* Michael Somos, Apr 21 2004 */
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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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