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%I #17 Mar 12 2021 22:24:43
%S 1,-1,0,-1,1,-2,2,-1,3,-3,3,-3,4,-5,5,-7,8,-8,9,-10,13,-15,14,-17,20,
%T -23,24,-26,31,-34,38,-41,46,-52,55,-62,70,-75,82,-90,103,-112,118,
%U -131,145,-161,172,-185,208,-225,244,-265,288,-316,339,-370,404,-435,469,-507,557,-601,640,-696,755,-818
%N McKay-Thompson series of class 40d for the Monster group.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H G. C. Greubel, <a href="/A112182/b112182.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</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 chi(-x) * chi(-x^5) in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Jul 02 2014
%F Expansion of q^(1/4) * eta(q) * eta(q^5) / (eta(q^2) * eta(q^10)) in powers of q. - _Michael Somos_, Jul 02 2014
%F Euler transform of period 10 sequence [ -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, ...]. - _Michael Somos_, Jul 02 2014
%F Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^3 - v) * (u-v^3) - 3 * u*v * (1 + u*v). - _Michael Somos_, Jul 02 2014
%F a(n) = (-1)^n * A112209(n).
%F a(n) ~ (-1)^n * exp(Pi*sqrt(n/5)) / (2^(3/2) * 5^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018
%e G.f. = 1 - x - x^3 + x^4 - 2*x^5 + 2*x^6 - x^7 + 3*x^8 - 3*x^9 + 3*x^10 + ...
%e T40d = 1/q - q^3 - q^11 + q^15 - 2*q^19 + 2*q^23 - q^27 + 3*q^31 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^5, x^10], {x, 0, n}]; (* _Michael Somos_, Jul 02 2014 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^5 + A) / (eta(x^2 + A) * eta(x^10 + A)), n))}; /* _Michael Somos_, Jul 02 2014 */
%Y Cf. A112209.
%K sign
%O 0,6
%A _Michael Somos_, Aug 28 2005
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