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%I #36 Feb 07 2023 12:25:00
%S 1,-1,-1,0,1,0,-1,1,2,-1,-2,1,2,-1,-3,1,4,-2,-5,2,5,-2,-6,3,8,-4,-9,4,
%T 10,-4,-12,6,15,-7,-17,7,19,-8,-22,10,26,-12,-30,13,33,-14,-38,17,45,
%U -21,-51,22,56,-24,-64,29,74,-33,-83,36,92,-40,-104,46,119,-53,-133,58
%N McKay-Thompson series of class 32e for the Monster group.
%C Number 4 of the 130 identities listed in Slater 1952. - _Michael Somos_, Aug 21 2015
%H Seiichi Manyama, <a href="/A082303/b082303.txt">Table of n, a(n) for n = 0..10000</a>
%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann., 318 (2000), 255-275. see page 274.
%H Lucy Joan Slater, <a href="http://plms.oxfordjournals.org/content/s2-54/1/147.extract">Further Identities of the Rogers-Ramanujan Type</a>, Proc. London Math. Soc., Series 2, vol.s2-54, no.2, pp.147-167, (1952).
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Euler transform of period 4 sequence [ -1, -1, -1, 0, ...].
%F Expansion of q^(1/8) * eta(q) / eta(q^4) in powers of q.
%F Given g.f. A(x), then B(q) = (A(q^8) / q)^8 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v + 16) * (u + 16) * u - v^2. - _Michael Somos_, Jan 09 2005
%F G.f.: Product_{k>0} (1 - x^k) / (1 - x^(4*k)).
%F a(n) = (-1)^n * A029838(n).
%F Convolution square is A082304.
%F G.f.: 2 - 2/(1+Q(0)), where Q(k)= 1 - x^(2*k+1) - x^(2*k+1)/(1 + x^(2*k+2) + x^(2*k+2)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, May 02 2013
%F G.f.: Sum_{k>=0} (-1)^k * q^k^2 * Product_{i=1..k} (1 + x^(2*i - 1)) / (1 - x^(4*i)). - _Michael Somos_, Aug 21 2015
%F a(n) = -(1/n)*Sum_{k=1..n} A046897(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017
%F abs(a(n)) ~ sqrt(sqrt(2) + (-1)^n) * exp(Pi*sqrt(n)/2^(3/2)) / (4*n^(3/4)). - _Vaclav Kotesovec_, Feb 07 2023
%e G.f. = 1 - x - x^2 + x^4 - x^6 + x^7 + 2*x^8 - x^9 - 2*x^10 + x^11 + 2*x^12 + ...
%e T32e = 1/q - q^7 - q^15 + q^31 - q^47 + q^55 + 2*q^63 - q^71 - 2*q^79 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] / QPochhammer[ x^4], {x, 0, n}]; (* _Michael Somos_, Aug 20 2014 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] / QPochhammer[ -x^2, x^2], {x, 0, n}]; (* _Michael Somos_, Aug 20 2014 *)
%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (16 (1 - m)/m)^(1/8), {q, 0, n - 1/8}]]; (* _Michael Somos_, Aug 20 2014 *)
%t a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 1, n, 2}] / Product[ 1 + x^k, {k, 2, n, 2}], {x, 0, n}]; (* _Michael Somos_, Aug 20 2014 *)
%t a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-x}, {-x^2}, x^2, x], {x, 0, n}]; (* _Michael Somos_, Aug 21 2015 *)
%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^k^2 QPochhammer[ -x, x^2, k] / QPochhammer[ x^4, x^4, k], {k, 0, Sqrt@n}], {x, 0, n}]]; (* _Michael Somos_, Aug 21 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^4 + A), n))};
%o (PARI) q='q+O('q^66); Vec( eta(q)/eta(q^4) ) \\ _Joerg Arndt_, Mar 25 2017
%Y Cf. A029838, A082304.
%K sign
%O 0,9
%A _Michael Somos_, Apr 08 2003
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