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A029841
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McKay-Thompson series of class 8E for the Monster group.
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15
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1, 4, 2, -8, -1, 20, -2, -40, 3, 72, 2, -128, -4, 220, -4, -360, 5, 576, 8, -904, -8, 1384, -10, -2088, 11, 3108, 12, -4552, -15, 6592, -18, -9448, 22, 13392, 26, -18816, -29, 26216, -34, -36224, 38, 49700, 42, -67728, -51, 91688
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OFFSET
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0,2
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COMMENTS
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A Hauptmodul for Gamma'_0(8).
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REFERENCES
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A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 380, Section 488.
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LINKS
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J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339. See page 336.
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FORMULA
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G.f.: ( Product_{k>0} (1 + q^(2*k - 1)) / (1 + q^(2*k)) )^4.
Expansion of q^(1/4) * (1 + k) / k^(1/2) in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus. - Michael Somos, Aug 01 2011
Expansion of q^(1/2) * 4 / k in powers of q where q is Jacobi's nome and k is the elliptic modulus. - Michael Somos, Aug 01 2011 and Feb 28 2012
Expansion of (phi(x) / psi(x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(1/2) * (eta(q^2)^3 / (eta(q) * eta(q^4)^2))^4 in powers of q. - Michael Somos, Aug 01 2011
Euler transform of period 4 sequence [4, -8, 4, 0, ...]. - Michael Somos, Mar 18 2004
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 16 + 8*v + v^2 - u^2*v. - Michael Somos, Mar 18 2004
G.f. A(q) satisfies A(q) = sqrt(A(q^2))+4*q/sqrt(A(q^2)). - Joerg Arndt, Aug 06 2011
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EXAMPLE
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G.f. = 1 + 4*x + 2*x^2 - 8*x^3 - x^4 + 20*x^5 - 2*x^6 - 40*x^7 + 3*x^8 + ...
T8E = 1/q + 4*q + 2*q^3 - 8*q^5 - q^7 + 20*q^9 - 2*q^11 - 40*q^13 + 3*x^15 + ...
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MATHEMATICA
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a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ 4 / Sqrt[m], {q, 0, n - 1/2}]]; (* Michael Somos, Aug 01 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ 2 (1 + Sqrt[m]) / m^(1/4), {q, 0, n/2 - 1/4}]]; (* Michael Somos, Aug 01 2011 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 / (QPochhammer[ x] QPochhammer[x^4]^2))^4, {x, 0, n}]; (* Michael Somos, Aug 20 2014 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A))^2)^4, n))};
(PARI) {a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = (4*x + A) / sqrt(A)); polcoeff(A, n))};
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CROSSREFS
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KEYWORD
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sign,easy,nice
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AUTHOR
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STATUS
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approved
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