|
| |
|
|
A007245
|
|
McKay-Thompson series of class 3C for the Monster group.
(Formerly M5423)
|
|
10
| |
|
|
1, 248, 4124, 34752, 213126, 1057504, 4530744, 17333248, 60655377, 197230000, 603096260, 1749556736, 4848776870, 12908659008, 33161242504, 82505707520, 199429765972, 469556091240, 1079330385764, 2426800117504, 5346409013164
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
|
|
|
REFERENCES
| J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 37.
T. Gannon, Postcards from the edge, or Snapshots of the theory of generalised Moonshine, arXiv:math/0109067.
G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
McKay, John; Strauss, Hubertus. The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra 18 (1990), no. 1, 253-278.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
T. Gannon, [math.QA/0402345] Monstrous Moonshine: The first twenty-five years.
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
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group
|
|
|
FORMULA
| In the notation of Gunning, Lect. on Modular Forms, pp. 53-54, expand E_2(z) / Delta(z)^(1/3).
Given g.f. A(x), then B(x) = A(x^3)/x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - 54000 + 495 * u*v - (u*v)^2. - Michael Somos Apr 29 2006
Expansion of (phi(-q)^8 - (2 * phi(-q) * phi(q))^4 + 16 * phi(q)^8) / f(-q)^8 in powers of q where phi(), f() are Ramanujan theta functions.
|
|
|
EXAMPLE
| T3C = 1/q + 248*q^2 + 4124*q^5 + 34752*q^8 + 213126*q^11 + 1057504*q^14 + ...
|
|
|
MATHEMATICA
| 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}]
(* From Jean-François Alcover, May 17 2011, after M. Somos *)
|
|
|
PROG
| (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 */
|
|
|
CROSSREFS
| Sequence in context: A135046 A027654 A003916 * A178967 A030062 A033555
Adjacent sequences: A007242 A007243 A007244 * A007246 A007247 A007248
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
