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McKay-Thompson series of class 3C for the Monster group.
(Formerly M5423)
30

%I M5423 #79 Mar 12 2021 22:24:41

%S 1,248,4124,34752,213126,1057504,4530744,17333248,60655377,197230000,

%T 603096260,1749556736,4848776870,12908659008,33161242504,82505707520,

%U 199429765972,469556091240,1079330385764,2426800117504,5346409013164

%N McKay-Thompson series of class 3C for the Monster group.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%D G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.

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

%H Seiichi Manyama, <a href="/A007245/b007245.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..50 from Vincenzo Librandi)

%H J. H. Conway and S. P. Norton, <a href="https://doi.org/10.1112/blms/11.3.308">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.

%H N. D. Elkies, <a href="http://www.math.harvard.edu/~elkies/modular.pdf">Elliptic and modular curves over finite fields and related computational issues</a>, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 37.

%H 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 T. Gannon, <a href="http://arxiv.org/abs/math/0109067">Postcards from the edge, or Snapshots of the theory of generalised Moonshine</a>, arXiv:math/0109067.

%H T. Gannon, <a href="http://arXiv.org/abs/math.QA/0402345">Monstrous Moonshine: The first twenty-five years</a> [math.QA/0402345].

%H Yang-Hui He, John McKay, <a href="http://arxiv.org/abs/1505.06742">Sporadic and Exceptional</a>, arXiv:1505.06742 [math.AG], 2015.

%H G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (<a href="http://www.math.ksu.edu/~gerald/papers/dr.pdf">pdf</a>, <a href="http://www.math.ksu.edu/~gerald/papers/dr.ps.gz">ps</a>). see p.78. Table 5.1, c=8

%H G. Hoehn, <a href="http://arxiv.org/abs/math/0701626">Conformal designs based on vertex operator algebras</a>, arXiv:math/0701626 [math.QA], 23 Jan 2007.

%H 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 Michael Somos, <a href="/A007191/a007191.pdf">Emails to N. J. A. Sloane, 1993</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F In the notation of Gunning, Lectures on Modular Forms, pp. 53-54, expand E_2(z) / Delta(z)^(1/3).

%F Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^3 + v^3 - 54000 + 495 * u*v - (u*v)^2. - _Michael Somos_, Apr 29 2006

%F Expansion of (phi(-x)^8 - (2 * phi(-x) * phi(x))^4 + 16 * phi(x)^8) / f(-x)^8 in powers of x where phi(), f() are Ramanujan theta functions.

%F Expansion of chi(-x)^8 + 256 * x / chi(-x)^16 in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Jun 15 2013

%F Expansion of q^(1/3) * (eta(q) / eta(q^2))^8 + 256 * (eta(q^2) / eta(q))^16 in powers of q. - _Michael Somos_, Jun 15 2013

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jun 15 2013

%F a(n) ~ exp(4*Pi*sqrt(n/3)) / (sqrt(2)*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Dec 04 2015

%F Convolution cube is A000521. (The modular j-function)- _Michael Somos_, Sep 30 2019

%e G.f. = 1 + 248*x + 4124*x^2 + 34752*x^3 + 213126*x^4 + 1057504*x^5 + 4530744*x^6 + ...

%e T3C = 1/q + 248*q^2 + 4124*q^5 + 34752*q^8 + 213126*q^11 + 1057504*q^14 + ...

%t n = 21; f[u_, v_] = u^3 + v^3 - 54000 + 495*u*v - (u*v)^2;

%t a[x_] = Sum[c[k] x^k, {k, 0, n}]; b[x_] = a[x^3]/x;

%t eq[1] = # == 0 & /@ CoefficientList[x^6 f[b[x], b[x^2]], x] // Union // Rest; s[1] = Solve[eq[1][[1]], c[0]] // Last; Do[eq[k] = Rest[eq[k-1]] /. s[k-1] ; s[k] = Solve[eq[k][[1]], c[k-1]] // Last, {k, 2, n}]; Table[c[k], {k, 0, n-1}] /. Flatten @ Table[s[k], {k, 1, n}]

%t (* _Jean-François Alcover_, May 17 2011, after _Michael Somos_ *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^8 + 256 q QPochhammer[ q, q^2]^-16, {q, 0, n}]; (* _Michael Somos_, Jun 15 2013 *)

%t CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24) / (256*QPochhammer[-1, x]^8), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Sep 23 2017 *)

%t eta[q_]:= q^(1/24)*QPochhammer[q]; nmax = 55; f1A := (eta[q]/eta[q^2] )^24*(1 + 256*(eta[q^2]/eta[q])^24)^3; a:= CoefficientList[Series[(q*f1A + O[q]^nmax)^(1/3), {q,0,50}], q]; Table[a[[n]], {n,1,50}] (* _G. C. Greubel_, May 09 2018 *)

%t a[ n_] := SeriesCoefficient[ With[ {m = InverseEllipticNomeQ[q]}, (1 + 14 m + m^2) / (1 - m) / (4 m (1 - m))^(1/3)] 4 q^(1/3), {q, 0, n}] // Simplify; (* _Michael Somos_, Sep 30 2019 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, n, 240 * sigma(k, 3) * x^k, 1 + x * O(x^n)) / eta(x + x * O(x^n))^8, n))}; /* _Michael Somos_, Apr 17 2004 */

%o (PARI) {a(n) = if( n<0, 0, polcoeff( (x * ellj( x + x^2 * O(x^n)))^(1/3), n))}; /* _Michael Somos_, May 26 2004 */

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^8 + 256 * x * (eta(x^2 + A) / eta(x + A))^16, n))}; /* _Michael Somos_, Jun 15 2013 */

%Y Cf. A000521.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_