OFFSET
-1,2
COMMENTS
(1 + 42x + 783x^2 + 8672x^3 + ...) is the convolution square of (1 + 21x + 171x^2 + 745x^3 + ...), where A007261 = (1, 21, 171, 745, 2418, ...). - Gary W. Adamson, Jul 21 2009
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. 39.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
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
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = -1..2000
T. Piezas III, 0013: Article 3 (Pi Formulas and the Monster Group)
Titus Piezas III, On Ramanujan's Other Pi Formulas
FORMULA
Expansion of Hauptmodul for X_0^{+}(3).
Expansion of (h + 27)^2 / h, where h = (eta(q) / eta(q^3))^12.
Expansion of 27 * (b(q)^3 + c(q)^3)^2 / (b(q) * c(q))^3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 16 2012
Expansion of (a(q) / (eta(q) * eta(q^3)))^6 in powers of q where a() is a cubic AGM theta function. - Michael Somos, Dec 01 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 01 2013
a(n) ~ exp(4*Pi*sqrt(n/3)) / (sqrt(2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 07 2015
EXAMPLE
G.f. = 1/q + 42 + 783*q + 8672*q^2 + 65367*q^3 + 371520*q^4 + 1741655*q^5 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1/q ((QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / ( QPochhammer[ q] QPochhammer[ q^3]^2))^6, {q, 0, n}]; (* Michael Somos, Dec 01 2013 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( (1 + 27 * x * A)^2 / A, n))}; /* Michael Somos, Jun 16 2012 */
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved