|
| |
|
|
A007259
|
|
Expansion of Product (1+q^m)^(-8); m=1..inf.
(Formerly M4504)
|
|
1
| |
|
|
1, -8, 28, -64, 134, -288, 568, -1024, 1809, -3152, 5316, -8704, 13990, -22208, 34696, -53248, 80724, -121240, 180068, -264448, 384940, -556064, 796760, -1132544, 1598789, -2243056, 3127360, -4333568, 5971922, -8188096, 11170160, -15163392, 20491033, -27572936
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(
q) (A010054), chi(q) (A000700).
McKay-Thompson series of class 6F for the Monster group.
|
|
|
REFERENCES
| T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 118, Problem 24.
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
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.
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).
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
| Expansion of chi(-q)^8 in powers of q where chi() is a Ramanujan theta function. - Michael Somos Aug 18 2007
Expansion of q^(-1/3) * (eta(q) / eta(q^2))^8 in powers of q. - Michael Somos Aug 18 2007
Euler transform of period 2 sequence [ -8, 0, ...]. - Michael Somos Aug 18 2007
Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = v^2 - u^2 * v - 16 * u. - Michael Somos Aug 18 2007
G.f. is a Fourier series which satisfies f(-1 / (2 t)) = 16 / f(t) where q = exp(2 pi i t). - Michael Somos Aug 18 2007
a(2*n) = A014705(n). a(2*n + 1) = -8 * A022573(n). a(n) = A007263(3*n - 1).
|
|
|
EXAMPLE
| T6F = 1/q - 8q^2 + 28q^5 - 64q^8 + 134q^11 - 288q^14 + 568q^17 + ...
|
|
|
MATHEMATICA
| a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^8, {q, 0, n}] (* Michael Somos Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - q^k, {k, 1, n, 2}]^8, {q, 0, n}] (* Michael Somos Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^-8, {q, 0, n}] (* Michael Somos Jul 11 2011 *)
|
|
|
PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x *O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^8, n))}
|
|
|
CROSSREFS
| Cf. A007263, A014705, A022573.
Sequence in context: A007331 A002408 A101127 * A134747 A083013 A028553
Adjacent sequences: A007256 A007257 A007258 * A007260 A007261 A007262
|
|
|
KEYWORD
| sign,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
