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%I #17 Mar 12 2021 22:24:45
%S 1,-1,-1,0,1,0,-1,0,3,0,-2,0,2,0,-5,0,6,0,-7,0,7,0,-9,0,12,0,-13,0,16,
%T 0,-20,0,25,0,-27,0,31,0,-38,0,44,0,-51,0,58,0,-69,0,80,0,-92,0,102,0,
%U -118,0,141,0,-157,0,177,0,-203,0,234,0,-261,0,292,0
%N McKay-Thompson series of class 28C for the Monster group with a(0) = -1.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A161970/b161970.txt">Table of n, a(n) for n = -1..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-1) * chi(-q) * chi(-q^2) * chi(-q^7) * chi(-q^14) in powers of q where chi() is a Ramanujan theta function.
%F Expansion of eta(q) * eta(q^7) / (eta(q^4) * eta(q^28)) in powers of q.
%F Euler transform of period 28 sequence [ -1, -1, -1, 0, -1, -1, -2, 0, -1, -1, -1, 0, -1, -2, -1, 0, -1, -1, -1, 0, -2, -1, -1, 0, -1, -1, -1, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (u + 2) * (v + 2) - v^2.
%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * v * (u + 2) * (v + 2) * (4 + u + v + u*v).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A123648.
%F a(n) = A230446(n) unless n=0. a(n) = -(-1)^n * A230446(n). a(2*n) = 0 unless n=0. a(2*n - 1) = A058608(n).
%F Convolution inverse is A123648.
%e G.f. = 1/q - 1 - q + q^3 - q^5 + 3*q^7 - 2*q^9 + 2*q^11 - 5*q^13 + 6*q^15 + ...
%t a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q] QPochhammer[ q^7] / (QPochhammer[ q^4] QPochhammer[ q^28]), {q, 0, n}]; (* _Michael Somos_, Oct 18 2013 *)
%t a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q, q^2] QPochhammer[ q^2, q^4] QPochhammer[ q^7, q^14] QPochhammer[ q^14, q^28], {q, 0, n}]; (* _Michael Somos_, Sep 06 2015 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x*O(x^n); polcoeff( eta(x + A) * eta(x^7 + A) / (eta(x^4 + A) * eta(x^28 + A)),n))};
%Y Cf. A058608, A123648, A230446.
%K sign
%O -1,9
%A _Michael Somos_, Jun 22 2009
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