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A010819 Expansion of Product_{k>=1} (1 - x^k)^11.

%I A010819 #25 Jul 08 2025 01:46:17
%S A010819 1,-11,44,-55,-110,374,-143,-462,55,495,1287,-2069,-902,1210,-275,
%T A010819 3795,-1507,-2431,-3575,-385,8690,-1661,1143,1265,-4290,-12716,2299,
%U A010819 11440,3905,8635,-10472,6105,-20548,-1540,8690,-24904,29634,25003,8470,-23320,-18183
%N A010819 Expansion of Product_{k>=1} (1 - x^k)^11.
%D A010819 Newman, Morris; 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 A010819 Seiichi Manyama, <a href="/A010819/b010819.txt">Table of n, a(n) for n = 0..10000</a>
%H A010819 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 A010819 <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F A010819 Expansion of q^(-11/24) * eta(q)^11 in powers of q. - _Michael Somos_, May 28 2013
%F A010819 a(n) == A010815(n) (mod 11). - _Michael Somos_, May 28 2013
%F A010819 a(0) = 1, a(n) = -(11/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 27 2017
%F A010819 G.f.: exp(-11*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 05 2018
%e A010819 1 - 11*x + 44*x^2 - 55*x^3 - 110*x^4 + 374*x^5 - 143*x^6 - 462*x^7 + ...
%e A010819 q^11 - 11*q^35 + 44*q^59 - 55*q^83 - 110*q^107 + 374*q^131 - 143*q^155 + ...
%t A010819 a[ n_] := SeriesCoefficient[ QPochhammer[ q]^11, {q, 0, n}] (* _Michael Somos_, May 28 2013 *)
%t A010819 a[ n_] := SeriesCoefficient[ Product[ 1 - q^k, {k, n}]^11, {q, 0, n}] (* _Michael Somos_, May 28 2013 *)
%o A010819 (PARI) {a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^11, n))} /* _Michael Somos_, May 28 2013 */
%Y A010819 Cf. A010815.
%K A010819 sign
%O A010819 0,2
%A A010819 _N. J. A. Sloane_