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URL: https://oeis.org/A007249

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A007249
McKay-Thompson series of class 4D for the Monster group.
9
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
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.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 2 * exp(-Pi / 2) * sqrt(2) = A388182. - Simon Plouffe, Sep 15 2025
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
Column k=12 of A286352.
Sequence in context: A045853 A277104 A014787 * A112142 A271870 A114243
KEYWORD
sign
STATUS
approved