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%I #35 Mar 12 2021 22:24:42
%S 1,-1,1,1,-1,0,1,-2,0,2,-3,1,4,-4,1,4,-6,1,5,-8,1,8,-10,2,11,-14,4,14,
%T -19,4,17,-24,4,23,-31,6,31,-40,9,38,-50,10,46,-63,11,60,-79,16,77,
%U -98,21,92,-122,24,112,-150,28,140,-183,36,173,-224,46,208,-273,54,249,-329,62,304,-396,78,370,-478,98
%N McKay-Thompson series of class 36B for the Monster group.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A062244/b062244.txt">Table of n, a(n) for n = 0..1000</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.
%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>
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of a Hauptmodul for Gamma'_0(18).
%F G.f.: Product_{k>0} (1 + x^(6*k - 3))^3 / (1 + x^(2*k - 1)). - _Michael Somos_, Mar 17 2004
%F Expansion of q^(1/3) * eta(q) * eta(q^4) * eta(q^6)^6 / (eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3) in powers of q.
%F Given g.f. A(x), then B(q) = A(q^3) /q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v^4 - u^3*v^3 - 3*u^2*v^2 + v*u^4 + 4*u*v - 2. - _Michael Somos_, Mar 17 2004
%F Expansion of chi(x^3)^3 / chi(x) = f(-x, x^2) / psi(-x^3) = phi(x^3) / f(x, x^5) in powers of x where phi(), psi(), chi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - _Michael Somos_, Aug 10 2017
%F Euler transform of period 12 sequence [-1, 1, 2, 0, -1, -2, -1, 0, 2, 1, -1, 0, ...]. - _Michael Somos_, Sep 17 2007
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132972.
%F a(n) = (-1)^n * A062242(n). a(2*n) = A132179(n). a(2*n + 1) = - A092848(n).
%F Convolution inverse of A128111.
%e G.f. = 1 - x + x^2 + x^3 - x^4 + x^6 - 2*x^7 + 2*x^9 - 3*x^10 + x^11 + ...
%e T36B = 1/q - q^2 + q^5 + q^8 - q^11 + q^17 - 2*q^20 + 2*q^26 - 3*q^29 + ...
%t a[ n_] := (-1)^n SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3]^3 / (QPochhammer[ x] QPochhammer[ x^6]^3), {x, 0, n}]; (* _Michael Somos_, Aug 20 2014 *)
%t A062244[n_] := SeriesCoefficient[QPochhammer[-q^3, q^6]^3/QPochhammer[-q, q^2], {q, 0, n}]; Table[A062244[n], {n,0,50}] (* _G. C. Greubel_, Aug 09 2017 *)
%t a[ n_] := SeriesCoefficient[ 2 x^(1/3) / (EllipticTheta[ 3, 0, x^(1/3)] / EllipticTheta[ 3, 0, x^3] - 1), {x, 0, n}]; (* _Michael Somos_, Aug 10 2017 *)
%t a[ n_] := SeriesCoefficient[ x^(1/3) (1 + EllipticTheta[2, Pi/4, x^(1/6)] / EllipticTheta[2, Pi/4, x^(3/2)]), {x, 0, n}]; (* _Michael Somos_, Aug 10 2017 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x^3]/(QPochhammer[ -x, x^6] QPochhammer[ -x^5, x^6] QPochhammer[ x^6]), {x, 0, n}]; (* _Michael Somos_, Aug 10 2017 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3), n))}; /* _Michael Somos_, Jan 09 2005 */
%Y Cf. A062242, A092848, A128111, A132179, A132972.
%K sign,easy
%O 0,8
%A _N. J. A. Sloane_, Jul 01 2001
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