A007261 McKay-Thompson series of class 6b for the Monster group.
1, 21, 171, 745, 2418, 7587, 20510, 51351, 122715, 277384, 598812, 1255761, 2543973, 5011725, 9653013, 18176040, 33535032, 60831648, 108490390, 190557015, 330174837, 564626278, 953857104, 1593681480, 2634409140, 4311592119, 6991502688, 11237020682, 17909802270
Offset: 0
Keywords
Examples
1 + 21*x + 171*x^2 + 745*x^3 + 2418*x^4 + 7587*x^5 + 20510*x^6 + 51351*x^7 + ... T6b = 1/q + 21*q + 171*q^3 + 745*q^5 + 2418*q^7 + 7587*q^9 + 20510*q^11 + ...
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..3000
- 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).
- 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.
- Index entries for McKay-Thompson series for Monster simple group
Crossrefs
Cf. A030197. - Gary W. Adamson, Jul 21 2009
Cf. A058537.
Programs
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Mathematica
a[0] = 1; a[n_] := Module[{A = x*O[x]^n}, A = (QPochhammer[x^3 + A] / QPochhammer[x + A])^12; SeriesCoefficient[Sqrt[(1 + 27*x*A)^2/A], n]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 06 2015, adapted from Michael Somos's PARI script *) CoefficientList[Series[(QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3)^3 / (QPochhammer[x, x]^3*QPochhammer[x^3, x^3]^6), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 07 2015 *) nmax = 30; CoefficientList[Series[Product[(1 - x^k)^6/(1 - x^(3*k))^6, {k, 1, nmax}] + 27*x*Product[(1 - x^(3*k))^6/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2017 *) -
PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( sqrt((1 + 27 * x * A)^2 / A), n))} /* Michael Somos, Jun 16 2012 */ -
PARI
N=66; q='q+O('q^N); t=(eta(q) / eta(q^3))^6; Vec(t + 27*q/t) \\ Joerg Arndt, Mar 11 2017
Formula
Expansion of (27 * x * (b(x)^3 + c(x)^3)^2 / (b(x) * c(x))^3)^(1/2) in powers of x where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 16 2012
Convolution cube of A058537. - Michael Somos, Aug 20 2012
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 07 2015
Expansion of q^(1/2) * (eta(q)^6/eta(q^3)^6 + 27*eta(q^3)^6/eta(q)^6) in powers of q. - G. A. Edgar, Mar 10 2017
Comments