A010827 - OEIS

%I #36 Feb 05 2018 16:35:45

%S 1,-21,189,-910,2205,-378,-13321,33345,-10395,-86870,122703,46683,

%T -98287,-264915,96390,1163064,-1113588,-1066527,1042055,536025,

%U 2287467,-3603805,-1391733,478170,-562555,13742379,-7889805

%N Expansion of Product_{k>=1} (1 - x^k)^21.

%C If not n == 0 (mod 7) then a(n) == 0 (mod 7). The reverse is not true, e.g., a(14) = 96390 == (0 mod 7). See the Hardy reference, p. 165. - _Wolfdieter Lang_, Jan 28 2017

%D G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 165.

%D 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 Seiichi Manyama, <a href="/A010827/b010827.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. MR1955423 (2003k:11071).

%H <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>

%F G.f.: Product_{k>0} (1 - x^k)^21.

%F a(0) = 1, a(n) = -(21/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 27 2017

%F G.f.: exp(-21*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 05 2018

%e G.f. = 1 - 21*x + 189*x^2 - 910*x^3 + 2205*x^4 - 378*x^5 - 13321*x^6 + 33345*x^7 + ...

%t CoefficientList[Expand@ Product[(1 - x^k)^21, {k, 27}], x, 27] (* _Michael De Vlieger_, Jun 08 2016 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^21, {x, 0, n}]; (* _Michael Somos_, Jan 28 2017 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^21, n))}; /* _Michael Somos_, Jan 28 2017 */

%Y Cf. A126581.

%K sign

%O 0,2

%A _N. J. A. Sloane_