A008705 Coefficient of x^n in (Product_{m=1..n}(1-x^m))^n.

1, -1, -1, 5, -5, -6, 11, 41, -125, -85, 1054, -2069, -209, 8605, -15625, 3990, 14035, 36685, -130525, -254525, 1899830, -3603805, -134905, 13479425, -25499225, 23579969, -64447293, 237487433, -133867445, -1795846200, 6309965146, -6788705842, -11762712973

Offset: 0

Author

T. Forbes (anthony.d.forbes(AT)googlemail.com)

Keywords

• sign

Comments

Degree of resulting polynomial is A002411(n). - Michel Marcus, Sep 05 2013

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k. - Peter Bala, Jan 31 2022

Conjectures: the supercongruences a(p) == -1 - p (mod p^2) and a(2*p) == p - 1 (mod p^2) hold for all primes p >= 3. - Peter Bala, Apr 18 2023

Examples

			(1-x)^1 = -x + 1, hence a(1) = -1.
(1-x^2)^2*(1-x)^2 = x^6 - 2*x^5 - x^4 + 4*x^3 - x^2 - 2*x + 1, hence a(2) = -1.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..2856 (terms 0..256 from N. J. A. Sloane)
• Morris Newman, Further identities and congruences for the coefficients of modular forms, Canadian J. Math 10 (1958): 577-586. See Table 1, column p=5.
• Morris Newman, Further identities and congruences for the coefficients of modular forms [annotated scanned copy], Canadian J. Math 10 (1958): 577-586. See Table 1, column p=5.

Crossrefs

Bisections: A262308, A262309.

Main diagonal of A286354.

Programs

Maple

C5:=proc(r) local t1,n; t1:=mul((1-x^n)^r,n=1..r+2); series(t1,x,r+1); coeff(%,x,r); end;
[seq(C5(i),i=0..30)]; # N. J. A. Sloane, Oct 04 2015
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, -k*
      add(numtheory[sigma](j)*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 21 2018

Mathematica

With[{m = 40}, Table[SeriesCoefficient[Series[(Product[1-x^j, {j, n}])^n, {x, 0, m}], n], {n, 0, m}]] (* G. C. Greubel, Sep 09 2019 *)

PARI

a(n) = polcoeff(prod(m = 1, n, (1-x^m)^n), n); \\ Michel Marcus, Sep 05 2013

Formula

a(n) = [x^n] exp(-n*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, May 30 2018

Extensions

More terms from Michel Marcus, Sep 05 2013

a(0)=1 prepended by N. J. A. Sloane, Oct 04 2015