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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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26
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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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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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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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Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..50 from Vincenzo Librandi)
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 37.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
T. Gannon, Postcards from the edge, or Snapshots of the theory of generalised Moonshine, arXiv:math/0109067.
T. Gannon, Monstrous Moonshine: The first twenty-five years [math.QA/0402345].
Yang-Hui He, John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
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
G. Hoehn, Conformal designs based on vertex operator algebras, arXiv:math/0701626 [math.QA], 23 Jan 2007.
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
M. Somos, Emails to N. J. A. Sloane, 1993
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to groups
Index entries for McKay-Thompson series for Monster simple group
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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
Convolution cube is A000521. (The modular j-function)- Michael Somos, Sep 30 2019
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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}]
(* Jean-François Alcover, May 17 2011, after Michael Somos *)
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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Cf. A000521.
Sequence in context: A135046 A027654 A003916 * A327959 A178967 A030062
Adjacent sequences: A007242 A007243 A007244 * A007246 A007247 A007248
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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