|
|
A008705
|
|
Coefficient of x^n in (Product_{m=1..n}(1-x^m))^n.
|
|
12
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
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
|
|
LINKS
|
|
|
FORMULA
|
a(n) = [x^n] exp(-n*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, May 30 2018
|
|
EXAMPLE
|
(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.
|
|
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;
# 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):
|
|
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 *)
|
|
PROG
|
(PARI) a(n) = polcoeff(prod(m = 1, n, (1-x^m)^n), n); \\ Michel Marcus, Sep 05 2013
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
T. Forbes (anthony.d.forbes(AT)googlemail.com)
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|