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A007249 McKay-Thompson series of class 4D for the Monster group.

%I A007249 M4846 #50 Feb 16 2025 08:32:31
%S A007249 1,-12,66,-232,639,-1596,3774,-8328,17283,-34520,66882,-125568,229244,
%T A007249 -409236,716412,-1231048,2079237,-3459264,5677832,-9200232,14729592,
%U A007249 -23325752,36567222,-56778888,87369483,-133315692
%N A007249 McKay-Thompson series of class 4D for the Monster group.
%C A007249 The convolution square root of A007191, and also the left and right borders of the triangle A161196. - _Gary W. Adamson_, Jun 06 2009
%C A007249 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A007249 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007249 Seiichi Manyama, <a href="/A007249/b007249.txt">Table of n, a(n) for n = 0..1000</a>
%H A007249 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 A007249 D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H A007249 J. McKay and H. 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 A007249 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A007249 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A007249 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A007249 G.f.: Product_{m>=1} (1 + x^m)^(-12).
%F A007249 Expansion of chi(-x)^12 in powers of x where chi() is a Ramanujan theta function.
%F A007249 G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 64 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022577. - _Michael Somos_, Jul 22 2011
%F A007249 a(n) = (-1)^n * A112142(n). (class 8B). Convolution inverse of A022577. - _Michael Somos_, Jul 22 2011
%F A007249 a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (2^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015
%F A007249 a(0) = 1, a(n) = -(12/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 05 2017
%F A007249 G.f.: exp(-12*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018
%F A007249 Expansion of q^(1/2)*(eta(q)/eta(q^2))^12 in powers of q. - _G. C. Greubel_, Feb 13 2018
%e A007249 1 - 12*x + 66*x^2 - 232*x^3 + 639*x^4 - 1596*x^5 + 3774*x^6 + ...
%e A007249 T4D = 1/q - 12*q + 66*q^3 - 232*q^5 + 639*q^7 - 1596*q^9 + 3774*q^11 - ...
%t A007249 a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m) / (m / 16 / q)^(1/2), {q, 0, n}]] (* _Michael Somos_, Jul 22 2011 *)
%t A007249 a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/2) / (m / 16 / q), {q, 0, 2 n}]] (* _Michael Somos_, Jul 22 2011 *)
%t A007249 nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^12, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 27 2015 *)
%t A007249 QP = QPochhammer; s = (QP[q]/QP[q^2])^12 + O[q]^30; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 12 2015, adapted from PARI *)
%t A007249 eta[q_]:=q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(1/2)*(eta[q]/eta[q^2])^12, {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Feb 13 2018 *)
%o A007249 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^12, n))} /* _Michael Somos_, Jul 22 2011 */
%Y A007249 Cf. A007191, A022577, A112142.
%Y A007249 Column k=12 of A286352.
%K A007249 sign
%O A007249 0,2
%A A007249 _N. J. A. Sloane_