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A007247 McKay-Thompson series of class 4B for the Monster group.
(Formerly M5305)
5
1, 52, 834, 4760, 24703, 94980, 343998, 1077496, 3222915, 8844712, 23381058, 58359168, 141244796, 327974700, 742169724, 1627202744, 3490345477, 7301071680, 14987511560, 30138820888, 59623576440, 115928963656 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..500

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 4 * q * (1 + k'^2)^2 / (k' * k^2) in powers of q^2 where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.

Expansion of 4 * q^(1/2) * (k'^4 + 4*k^2) / (k'^2 * k) in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 22 2011

a(n) = A007249(n) + 64 * A022577(n - 1). - Michael Somos, Jul 22 2011

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

EXAMPLE

T4B = 1/q + 52*q + 834*q^3 + 4760*q^5 + 24703*q^7 + 94980*q^9 + ...

MATHEMATICA

a[ n_] := Module[ {m = InverseEllipticNomeQ @ q, e}, e = (1 - m) / (m / 16)^(1/2); SeriesCoefficient[ (e + 64 / e), {q, 0, n - 1/2}]] (* Michael Somos, Jul 11 2011 *)

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ 4 (2 - m)^2 / (m (1 - m)^(1/2)), {q, 0, 2 n - 1}]] (* Michael Somos, Jul 22 2011 *)

QP = QPochhammer; A = (QP[q]/QP[q^2])^12; s = A + 64*(q/A) + O[q]^30; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 15 2015, adapted from 2nd PARI script *)

nmax = 30; CoefficientList[Series[64*x*Product[(1 + x^k)^12, {k, 1, nmax}] + Product[1/(1 + x^k)^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 01 2017 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = prod( k=1, (n+1)\2, 1 - x^(2*k - 1), 1 + x * O(x^n))^12; polcoeff( A + 64 * x / A, n))} /* Michael Somos, Jul 22 2011 */

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( A + 64 * x / A, n))} /* Michael Somos, Nov 11 2006 */

(PARI) { my(q='q+O('q^66), t=(eta(q)/eta(q^2))^12); Vec( t + 64*q/t ) } \\ Joerg Arndt, Apr 02 2017

CROSSREFS

Cf. A007249, A022577.

Sequence in context: A264309 A160344 A163691 * A232312 A278001 A083936

Adjacent sequences:  A007244 A007245 A007246 * A007248 A007249 A007250

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 11 21:15 EST 2017. Contains 295919 sequences.