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A007245
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McKay-Thompson series of class 3C for the Monster group.
(Formerly M5423)
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30
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1, 248, 4124, 34752, 213126, 1057504, 4530744, 17333248, 60655377, 197230000, 603096260, 1749556736, 4848776870, 12908659008, 33161242504, 82505707520, 199429765972, 469556091240, 1079330385764, 2426800117504, 5346409013164
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OFFSET
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0,2
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COMMENTS
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REFERENCES
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G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps). see p.78. Table 5.1, c=8
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FORMULA
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In the notation of Gunning, Lectures on Modular Forms, pp. 53-54, expand E_2(z) / Delta(z)^(1/3).
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^3 + v^3 - 54000 + 495 * u*v - (u*v)^2. - Michael Somos, Apr 29 2006
Expansion of (phi(-x)^8 - (2 * phi(-x) * phi(x))^4 + 16 * phi(x)^8) / f(-x)^8 in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of chi(-x)^8 + 256 * x / chi(-x)^16 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Jun 15 2013
Expansion of q^(1/3) * (eta(q) / eta(q^2))^8 + 256 * (eta(q^2) / eta(q))^16 in powers of q. - Michael Somos, Jun 15 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 15 2013
a(n) ~ exp(4*Pi*sqrt(n/3)) / (sqrt(2)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Dec 04 2015
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EXAMPLE
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G.f. = 1 + 248*x + 4124*x^2 + 34752*x^3 + 213126*x^4 + 1057504*x^5 + 4530744*x^6 + ...
T3C = 1/q + 248*q^2 + 4124*q^5 + 34752*q^8 + 213126*q^11 + 1057504*q^14 + ...
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MATHEMATICA
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n = 21; f[u_, v_] = u^3 + v^3 - 54000 + 495*u*v - (u*v)^2;
a[x_] = Sum[c[k] x^k, {k, 0, n}]; b[x_] = a[x^3]/x;
eq[1] = # == 0 & /@ CoefficientList[x^6 f[b[x], b[x^2]], x] // Union // Rest; s[1] = Solve[eq[1][[1]], c[0]] // Last; Do[eq[k] = Rest[eq[k-1]] /. s[k-1] ; s[k] = Solve[eq[k][[1]], c[k-1]] // Last, {k, 2, n}]; Table[c[k], {k, 0, n-1}] /. Flatten @ Table[s[k], {k, 1, n}]
a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^8 + 256 q QPochhammer[ q, q^2]^-16, {q, 0, n}]; (* Michael Somos, Jun 15 2013 *)
CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24) / (256*QPochhammer[-1, x]^8), {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; nmax = 55; f1A := (eta[q]/eta[q^2] )^24*(1 + 256*(eta[q^2]/eta[q])^24)^3; a:= CoefficientList[Series[(q*f1A + O[q]^nmax)^(1/3), {q, 0, 50}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, May 09 2018 *)
a[ n_] := SeriesCoefficient[ With[ {m = InverseEllipticNomeQ[q]}, (1 + 14 m + m^2) / (1 - m) / (4 m (1 - m))^(1/3)] 4 q^(1/3), {q, 0, n}] // Simplify; (* Michael Somos, Sep 30 2019 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, n, 240 * sigma(k, 3) * x^k, 1 + x * O(x^n)) / eta(x + x * O(x^n))^8, n))}; /* Michael Somos, Apr 17 2004 */
(PARI) {a(n) = if( n<0, 0, polcoeff( (x * ellj( x + x^2 * O(x^n)))^(1/3), n))}; /* Michael Somos, May 26 2004 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^8 + 256 * x * (eta(x^2 + A) / eta(x + A))^16, n))}; /* Michael Somos, Jun 15 2013 */
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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