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

1, -12, 66, -232, 639, -1596, 3774, -8328, 17283, -34520, 66882, -125568, 229244, -409236, 716412, -1231048, 2079237, -3459264, 5677832, -9200232, 14729592, -23325752, 36567222, -56778888, 87369483, -133315692

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Comments

The convolution square root of A007191, and also the left and right borders of the triangle A161196. - Gary W. Adamson, Jun 06 2009

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

Examples

			1 - 12*x + 66*x^2 - 232*x^3 + 639*x^4 - 1596*x^5 + 3774*x^6 + ...
T4D = 1/q - 12*q + 66*q^3 - 232*q^5 + 639*q^7 - 1596*q^9 + 3774*q^11 - ...
		

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• 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.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• Index entries for McKay-Thompson series for Monster simple group

Crossrefs

Cf. A007191, A022577, A112142.

Column k=12 of A286352.

Programs

Mathematica

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m) / (m / 16 / q)^(1/2), {q, 0, n}]] (* Michael Somos, Jul 22 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/2) / (m / 16 / q), {q, 0, 2 n}]] (* Michael Somos, Jul 22 2011 *)
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
QP = QPochhammer; s = (QP[q]/QP[q^2])^12 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015, adapted from PARI *)
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 *)

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 */

Formula

G.f.: Product_{m>=1} (1 + x^m)^(-12).

Expansion of chi(-x)^12 in powers of x where chi() is a Ramanujan theta function.

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

a(n) = (-1)^n * A112142(n). (class 8B). Convolution inverse of A022577. - Michael Somos, Jul 22 2011

a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (2^(5/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015

a(0) = 1, a(n) = -(12/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017

G.f.: exp(-12*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

Expansion of q^(1/2)*(eta(q)/eta(q^2))^12 in powers of q. - G. C. Greubel, Feb 13 2018