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A112142 McKay-Thompson series of class 8B for the Monster group. 5
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, 201825420, 303257512 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1000

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

M. 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

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

Expansion of q^(1/2) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^12 in powers of q.

G.f.: Product_{k>0} (1 + (-x)^k)^-12 = Product_{k>0} (1 + x^(2*k - 1))^-12.

a(n) = (-1)^n * A007249(n). Convolution inverse of A124863.

G.f.: T(0), where T(k) = 1 - 1/(1 - 1/(1 - 1/(1+(x)^(2*k+1))^12/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013

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

G.f.: exp(12*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018

EXAMPLE

1 + 12*x + 66*x^2 + 232*x^3 + 639*x^4 + 1596*x^5 + 3774*x^6 + 8328*x^7 + ...

T8B = 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_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}]^-12, {x, 0, n}] (* Michael Somos, Jul 22 2011 *)

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^12, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

QP = QPochhammer; s = (QP[q^2]^2/(QP[q]*QP[q^4]))^12 + O[q]^50; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 16 2015, adapted from PARI *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^12, n))}

CROSSREFS

Cf. A007249, A124863.

Sequence in context: A277104 A014787 A007249 * A271870 A114243 A000972

Adjacent sequences:  A112139 A112140 A112141 * A112143 A112144 A112145

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 28 2005

STATUS

approved

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Last modified July 16 21:48 EDT 2018. Contains 312693 sequences. (Running on oeis4.)