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A007245 McKay-Thompson series of class 3C for the Monster group.
(Formerly M5423)
11
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; text; internal format)
OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (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.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..50

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

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.

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

EXAMPLE

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 + ...

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}]

(* Jean-Fran├žois Alcover, May 17 2011, after M. Somos *)

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^8 + 256 q QPochhammer[ q, q^2]^-16, {q, 0, n}] (* Michael Somos, Jun 15 2013 *)

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 */

(PARI) {a(n) = local(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 */

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.

STATUS

approved

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Last modified July 7 06:29 EDT 2015. Contains 259313 sequences.