A010827 Expansion of Product_{k>=1} (1 - x^k)^21.

1, -21, 189, -910, 2205, -378, -13321, 33345, -10395, -86870, 122703, 46683, -98287, -264915, 96390, 1163064, -1113588, -1066527, 1042055, 536025, 2287467, -3603805, -1391733, 478170, -562555, 13742379, -7889805

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Comments

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

Examples

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

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071).
• Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Crossrefs

Cf. A126581.

Programs

Mathematica

CoefficientList[Expand@ Product[(1 - x^k)^21, {k, 27}], x, 27] (* Michael De Vlieger, Jun 08 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^21, {x, 0, n}]; (* Michael Somos, Jan 28 2017 *)

PARI

{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^21, n))}; /* Michael Somos, Jan 28 2017 */

Formula

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

a(0) = 1, a(n) = -(21/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017

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