A007249 McKay-Thompson series of class 4D for the Monster group. (Formerly M4846)

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

0,2

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).

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

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

Example

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 - ...

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 *)

Prog

(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 */

Crossrefs

Cf. A007191, A022577, A112142.

Column k=12 of A286352.

Sequence in context: A045853 A277104 A014787 * A112142 A271870 A114243

Adjacent sequences: A007246 A007247 A007248 * A007250 A007251 A007252

Keyword

sign

Author

N. J. A. Sloane

Status

approved