OFFSET
0,4
REFERENCES
T. Gannon, Moonshine Beyond the Monster, Cambridge, 2006; see p. 423.
G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
LINKS
FORMULA
Let X=sqrt( Sum (q^(m^2/2), m=-inf..inf) / q^(1/24) Product( 1-q^n, n=1..inf ) ). Then series is X^47 - 47*X^23.
a(n) ~ 47^(1/4) * exp(Pi*sqrt(47*n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Feb 05 2020
EXAMPLE
G.f. = 1 + 4371*x^3 + 96256*x^4 + 1143745*x^5 + 9646891*x^6 + 64680601*x^7 + ...
G.f. = q^(-47/48) * (1 + 4371 * q^(3/2) + 96256 * q^2 + 1143745 * q^(5/2) + ...).
MATHEMATICA
nmax = 30; CoefficientList[Series[Product[(1 + x^(2*k + 1))^47, {k, 0, nmax}] - 47*x*Product[(1 + x^(2*k + 1))^23, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 05 2020 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = prod( i=1, (1+n)\2, 1 + x^(2*i-1), 1 + x * O(x^n)); polcoeff( A^47 - 47 * x * A^23, n))}; /* Michael Somos, Jul 01 2004 */
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved