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%I #41 Aug 13 2023 08:47:38
%S 1,-48,1080,-15040,143820,-985824,4857920,-16295040,28412910,38671600,
%T -424520544,1268350272,-1211937160,-4306546080,18293091840,
%U -23522231424,-26299018683,137218594320,-150999182320,-134713340160
%N Expansion of Product_{k >= 1} (1-x^k)^48.
%D Morris Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
%H Seiichi Manyama, <a href="/A010839/b010839.txt">Table of n, a(n) for n = 0..10000</a>
%H M. Boylan, <a href="http://dx.doi.org/10.1016/S0022-314X(02)00037-9">Exceptional congruences for the coefficients of certain eta-product newforms</a>, J. Number Theory 98 (2003), no. 2, 377-389.
%H <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F Let b(q) be the determinant of the 3 X 3 Hankel matrix [E_4, E_6, E_8 ; E_6, E_8, E_10 ; E_8, E_10, E_12]. G.f. is -691*b(q)/(q^2*1728^2*250). - _Seiichi Manyama_, Jul 17 2017
%F a(n) = (A290010(n+2) - A290009(n+2) + 691*(A282330(n+2) - A282332(n+2)))/(1728^2*250). - _Seiichi Manyama_, Jul 19 2017
%F a(0) = 1, a(n) = -(48/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Aug 13 2023
%e 1 - 48*x + 1080*x^2 - 15040*x^3 + 143820*x^4 - 985824*x^5 + 4857920*x^6 - 16295040*x^7 + ...
%Y Column k=48 of A286354.
%Y Cf. A000203, A082558, A126581, A282330 (E_8^3), A282332 (E_6*E_8*E_10 = E4*E_10^2), A290009, A290010.
%K sign
%O 0,2
%A _N. J. A. Sloane_
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