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%I #23 Sep 07 2017 05:26:03
%S 1,0,1,-1,2,-2,2,-3,5,-5,5,-7,9,-10,11,-14,18,-20,22,-27,32,-36,40,
%T -48,57,-63,70,-82,95,-106,119,-137,158,-175,195,-222,252,-280,311,
%U -352,397,-439,486,-546,611,-676,747,-834,929,-1024,1128,-1253,1389,-1528
%N McKay-Thompson series of class 30G for the Monster group.
%D D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H Seiichi Manyama, <a href="/A058618/b058618.txt">Table of n, a(n) for n = -1..10000</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Euler transform of period 30 sequence [ 0, 1, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, -1, 1, 0, 0, ...]. - _Michael Somos_, Apr 06 2012
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A094023.
%F G.f.: (x * Product_{k>0} (1 + x^k) * (1 + x^(15*k)) * P(15, x^k))^(-1) where P(n, x) is the n-th cyclotomic polynomial.
%F Convolution inverse of A094022. a(2*n) = -A123630(n).
%F a(2*n) = -A094023(n) if n>0. - _Michael Somos_, Aug 26 2015
%F a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017
%e T30G = 1/q + q - q^2 + 2*q^3 - 2*q^4 + 2*q^5 - 3*q^6 + 5*q^7 - 5*q^8 + ...
%t a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q^3] QPochhammer[ q^5] / (QPochhammer[ q^2] QPochhammer[ q^30]), {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^5 + A) / (eta(x^2 + A) * eta(x^30 + A)), n))}; /* _Michael Somos_, Apr 06 2012 */
%Y Cf. A094022, A094023, A123630.
%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K sign
%O -1,5
%A _N. J. A. Sloane_, Nov 27 2000
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