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A007241 McKay-Thompson series of class 2A for the Monster group with a(0) = 24.
(Formerly M5176)
197
1, 24, 4372, 96256, 1240002, 10698752, 74428120, 431529984, 2206741887, 10117578752, 42616961892, 166564106240, 611800208702, 2125795885056, 7040425608760, 22327393665024, 68134255043715, 200740384538624 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,2

REFERENCES

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 195.

S. Ramanujan, Modular Equations and Approximations to pi, pp. 23-39 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 26.

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..1000 from T. D. Noe)

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).

Yang-Hui He, John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.

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 sequences related to groups

Index entries for McKay-Thompson series for Monster simple group

FORMULA

G.f. 48 + 64(g_n^(24) + g_n^(-24)) where q = e^(-Pi sqrt(n)) and g_n is Ramanujan's class invariant. - Michael Somos, Apr 20 2005

a(n) ~ exp(2*Pi*sqrt(2*n)) / (2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Apr 01 2017

EXAMPLE

G.f. = 1/q + 24 + 4372*q + 96256*q^2 + 1240002*q^3 + 10698752*q^4 + ...

MATHEMATICA

a[n0_] := Module[{n=n0, A}, If[n < -1, 0, n++; A = Product[ 1 - x^(2*k-1) , {k, 1, Quotient[n+1, 2]}]^24; SeriesCoefficient[ A + x*48 + x^2*4096/A, {x, 0, n}]]]; Table[ a[n], {n, -1, 16}] (* Jean-Fran├žois Alcover, Oct 16 2012, after Michael Somos *)

a[ n_] := SeriesCoefficient[ With[{A = q QPochhammer[ -q, q]^24}, -80 + (1 + 64 A)^2 / A], {q, 0, n}]; (* Michael Somos, Apr 06 2015 *)

nmax = 50; CoefficientList[Series[48*x + 4096*x^2*Product[(1 + x^k)^24, {k, 1, nmax}] + Product[1/(1 + x^k)^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 01 2017 *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = prod( k=1, (n+1)\2, 1 - x^(2*k - 1), 1 + x * O(x^n))^24; polcoeff( A + x*48 + x^2 * 4096/A, n))}; /* Michael Somos, Feb 07 2003 */

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^2 + A) / eta(x + A))^24; polcoeff( -80*x + (1 + 64 * x * A)^2 / A, n))}; /* Michael Somos, Apr 06 2015 */

CROSSREFS

A045478, A007241, A106207, A007267, A101558 are all essentially the same sequence.

Sequence in context: A159399 A184687 * A106207 A100089 A151598 A003787

Adjacent sequences:  A007238 A007239 A007240 * A007242 A007243 A007244

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 12 06:05 EST 2017. Contains 295937 sequences.