|
| |
|
|
A022597
|
|
Expansion of Product (1 + q^m)^(-2); m=1..inf.
|
|
3
| |
|
|
1, -2, 1, -2, 4, -4, 5, -6, 9, -12, 13, -16, 21, -26, 29, -36, 46, -54, 62, -74, 90, -106, 122, -142, 171, -200, 227, -264, 311, -358, 408, -470, 545, -626, 709, -810, 933, -1062, 1198, -1362, 1555, -1760, 1980, -2238, 2536, -2858, 3205, -3602, 4063, -4560, 5092, -5704, 6400, -7150, 7966
(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 24J for the Monster group.
|
|
|
REFERENCES
| T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^2.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
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
D. Foata and G.-N. Han, Jacobi and Watson Identities Combinatorially Revisited
|
|
|
FORMULA
| Expansion of q^(1/12)(eta(q) / eta(q^2))^2 in powers of q.
Euler transform of period 2 sequence [ -2, 0, ...]. - Michael Somos Sep 10 2005
Expansion of chi(-q)^2 in powers of q where chi() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 2 / f(t) where q = exp(2 pi i t).
G.f.: Product_{k>0} (1 + x^k)^-2.Convolution square of A081362. Convolution inverse of A022567. a(n) = (-1)^n * A073252(n).
|
|
|
EXAMPLE
| T24J = 1/q - 2*q^11 + q^23 - 2*q^35 + 4*q^47 - 4*q^59 + 5*q^71 - 6*q^83 + ...
|
|
|
MATHEMATICA
| a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^2 , {q, 0, n}] (* Michael Somos Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^-2, {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))^2, n))} /* Michael Somos Sep 10 2005 */
|
|
|
CROSSREFS
| Cf. A022567, A073252, A081362.
Sequence in context: A023673 A132965 * A073252 A134005 A132320 A076369
Adjacent sequences: A022594 A022595 A022596 * A022598 A022599 A022600
|
|
|
KEYWORD
| sign,nice,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
