OFFSET
-1,2
COMMENTS
McKay-Thompson series of class 2A for the Monster group with a(0) = 104.
REFERENCES
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 195.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 517.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..1000 from T. D. Noe)
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
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.
Titus Piezas III, Ramanujan's Constant exp(Pi sqrt(163)) And Its Cousins.
Michael Somos, Emails to N. J. A. Sloane, 1993
FORMULA
Expansion of 16 * (1 + k'^2)^4 /(k' * k^2)^2 in powers of q^2. - Michael Somos, Nov 11 2006
a(n) ~ exp(2*Pi*sqrt(2*n)) / (2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Apr 01 2017
EXAMPLE
G.f. = 1/q + 104 + 4372*q + 96256*q^2 + 1240002*q^3 + 10698752*q^4 + ...
MATHEMATICA
a[ n_] := If[ n < -1, 0, With[ {m = InverseEllipticNomeQ[ q]}, SeriesCoefficient[ 16 (1 + m)^4 /(m (1 - m)^2), {q, 0, n}]]]; (* Michael Somos, Jun 29 2011 *)
a[ n_] := If[ n < -1, 0, With[ {m = ModularLambda[ Log[q]/(Pi I)]}, SeriesCoefficient[ 16 (1 + m)^4 /(m (1 - m)^2), {q, 0, n}]]]; (* Michael Somos, Jun 30 2011 *)
QP = QPochhammer; A = (QP[q]/QP[q^2])^12; s = (A + 64*(q/A))^2 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)
nmax = 20; CoefficientList[Series[128*x + Product[1/(1 + x^k)^24, {k, 1, nmax}] + 4096*x^2*Product[(1 + x^k)^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 03 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, A = prod(k=1, n\2 + 1, 1 - x^(2*k - 1), 1 + x^2 * O(x^n))^12; polcoeff( (64 * x / A + A)^2, n+1))};
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( (A + 64 * x / A)^2, n))}; /* Michael Somos, Nov 11 2006 */
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, Apr 28 1994
STATUS
approved