A007247 McKay-Thompson series of class 4B for the Monster group.

1, 52, 834, 4760, 24703, 94980, 343998, 1077496, 3222915, 8844712, 23381058, 58359168, 141244796, 327974700, 742169724, 1627202744, 3490345477, 7301071680, 14987511560, 30138820888, 59623576440, 115928963656

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

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

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000 (terms 0..500 from Vincenzo Librandi)
• J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
• Claude Duhr and Sara Maggio, Feynman integrals, elliptic integrals and two-parameter K3 surfaces, arXiv:2502.15326 [hep-th], 2025. See p. 12.
• David Ford, John McKay, and Simon Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
• John McKay and Hubertus 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. A007249, A022577.

Programs

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-François 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 *)

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

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