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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 (list; graph; refs; listen; history; text; internal format)
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

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

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-Fran├žois 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

Adjacent sequences:  A007258 A007259 A007260 * A007262 A007263 A007264

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 12 00:07 EST 2018. Contains 318052 sequences. (Running on oeis4.)