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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). Eric Weisstein's World of Mathematics, Ramanujan Theta Functions 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 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-François 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 December 10 20:48 EST 2017. Contains 295856 sequences.