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A007261
McKay-Thompson series of class 6b for the Monster group.
(Formerly M5111)
2
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
0,2
COMMENTS
From Gary W. Adamson, Jul 21 2009: (Start)
(1 + 21x + 171x^2 + 745x^3 + ...)^2 = (1 + 42x + 783x^2 + 8672x^3 + ...)
where A030197 = (1, 42, 783, 8672, 65367, ...). (End)
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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.
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
a(n) = A007262(n) + 27 * A121596(n-1). - Sean A. Irvine, Nov 26 2017
EXAMPLE
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 + ...
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 *)
PROG
(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
CROSSREFS
Cf. A030197. - Gary W. Adamson, Jul 21 2009
Cf. A058537.
Sequence in context: A126516 A241219 A185128 * A119105 A015880 A113163
KEYWORD
nonn
STATUS
approved