A007260 - OEIS

%I M5238 #40 Feb 14 2021 01:17:26

%S 1,-33,-153,-713,-2550,-7479,-20314,-51951,-122229,-276656,-601068,

%T -1254105,-2541531,-5018721,-9647991,-18168984,-33554784,-60818040,

%U -108471674,-190607871,-330140403,-564580142,-953980392,-1593599832,-2634301308,-4311874755,-6991318008

%N McKay-Thompson series of class 6a for Monster.

%C A more correct name would be: Expansion of replicable function of class 6a. See Alexander et al., 1992. - _N. J. A. Sloane_, Jun 12 2015

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vaclav Kotesovec, <a href="/A007260/b007260.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..499 from G. A. Edgar)

%H D. Alexander, C. Cummins, J. McKay and C. Simons, <a href="/A007242/a007242_1.pdf">Completely replicable functions</a>, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.

%H J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%H Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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.]

%H J. McKay and H. Strauss, <a href="http://dx.doi.org/10.1080/00927879008823911">The q-series of monstrous moonshine and the decomposition of the head characters</a>, Comm. Algebra 18 (1990), no. 1, 253-278.

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F Expansion of q * ((eta(q^2) / eta(q^6))^6 - 27 * (eta(q^6) / eta(q^2))^6) in powers of q^2. - _Michael Somos_, Jun 14 2015

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = -1 / f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jun 14 2015

%F a(n) = A007262(n) - 27 * A121596(n-1). - _Michael Somos_, Jun 14 2015

%F Convolution square is A258917. - _Michael Somos_, Jun 14 2015

%F a(n) ~ -exp(2*Pi*sqrt(2*n/3)) / (2^(3/4)*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2017

%e G.f. = 1 - 33*x - 153*x^2 - 713*x^3 - 2550*x^4 - 7479*x^5 - 20314*x^6 + ...

%e T6a = 1/q - 33*q - 153*q^3 - 713*q^5 - 2550*q^7 - 7479*q^9 - 20314*q^11 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^3])^6 - 27 x (QPochhammer[ x^3] / QPochhammer[ x])^6, {x, 0, n} ]; (* _Michael Somos_, Jun 14 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^3 + A))^6 - 27 * x * (eta(x^3 + A) / eta(x + A))^6, n))}; /* _Michael Somos_, Jun 14 2015 */

%o (PARI) { my(q='q+O('q^66), t=(eta(q)/eta(q^3))^6 ); Vec( t - 27*q/t ) } \\ _Joerg Arndt_, Apr 02 2017

%Y Cf. A007242, A007250, A007262, A121596, A258917.

%K sign

%O 0,2

%A _N. J. A. Sloane_.