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

%I A007247 M5305 #47 Feb 28 2025 09:37:52
%S A007247 1,52,834,4760,24703,94980,343998,1077496,3222915,8844712,23381058,
%T A007247 58359168,141244796,327974700,742169724,1627202744,3490345477,
%U A007247 7301071680,14987511560,30138820888,59623576440,115928963656
%N A007247 McKay-Thompson series of class 4B for the Monster group.
%D A007247 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007247 G. C. Greubel, <a href="/A007247/b007247.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..500 from Vincenzo Librandi)
%H A007247 J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.
%H A007247 Claude Duhr and Sara Maggio, <a href="https://arxiv.org/abs/2502.15326">Feynman integrals, elliptic integrals and two-parameter K3 surfaces</a>, arXiv:2502.15326 [hep-th], 2025. See p. 12.
%H A007247 David Ford, John McKay, and Simon Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H A007247 John McKay and Hubertus Strauss, <a href="http://dx.doi.org/10.1080/00927879008823911">The q-series of monstrous moonshine and the decomposition of the head characters</a>, Comm. Algebra 18 (1990), no. 1, 253-278.
%H A007247 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A007247 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.
%F A007247 Expansion of 4 * q^(1/2) * (k'^4 + 4*k^2) / (k'^2 * k) in powers of q.
%F A007247 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
%F A007247 a(n) = A007249(n) + 64 * A022577(n - 1). - _Michael Somos_, Jul 22 2011
%F A007247 a(n) ~ exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - _Vaclav Kotesovec_, Apr 01 2017
%e A007247 T4B = 1/q + 52*q + 834*q^3 + 4760*q^5 + 24703*q^7 + 94980*q^9 + ...
%t A007247 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 *)
%t A007247 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 *)
%t A007247 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 *)
%t A007247 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 *)
%o A007247 (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 */
%o A007247 (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 */
%o A007247 (PARI) { my(q='q+O('q^66), t=(eta(q)/eta(q^2))^12); Vec( t + 64*q/t ) } \\ _Joerg Arndt_, Apr 02 2017
%Y A007247 Cf. A007249, A022577.
%K A007247 nonn
%O A007247 0,2
%A A007247 _N. J. A. Sloane_