OFFSET
2,2
LINKS
So Okada, Homological mirror symmetry of Fermat polynomials, arxiv:0910.2014 [math.AG], 2009-2010.
FORMULA
The formula given by Okada needs to be corrected as follows:
Sum_{j=0..n-1} Sum_{i=0..n-1-j} n^j * binomial(n,j) * (-1)^(i+n+j) * binomial(n-2-j+1,i+1) * q^i.
From Peter Luschny, Jan 26 2017: (Start)
T(n,k) = [x^k] Sum_{j=0..n-1} t(j, n) for n>=2 and 0<=k<=n-2 with t(j,n) = (-1)^(j+n)*binomial(n,j)*(1-(1-x)^(n-1-j))*x^(-1)*n^j.
T(n,k) = [x^k] ((-n-x+1)^n+(x-1)*(1-n)^n-(-n)^n*x)*(-1)^n/((x-1)*x). (End)
EXAMPLE
The first few polynomials are 1; q + 7; q^2 + 13*q + 67; ...
Triangle begins:
1;
7, 1;
67, 13, 1;
821, 181, 21, 1;
12281, 2906, 406, 31, 1;
217015, 53719, 8359, 799, 43, 1;
4424071, 1129899, 188707, 20637, 1429, 57, 1;
...
MAPLE
T:= n-> (p-> seq(coeff(p, q, i), i=0..n-2))(add(add(n^j*
binomial(n, j)*(-1)^(i+n+j)*binomial(n-2-j+1, i+1)*
q^i, i=0..n-1-j), j=0..n-1)):
seq(T(n), n=2..10); # Alois P. Heinz, Jan 25 2017
# Alternatively:
t := n -> factor(((-n-x+1)^n+(x-1)*(1-n)^n-(-n)^n*x)*(-1)^n/((x-1)*x)):
seq(seq(coeff(t(n), x, k), k=0..n-2), n=2..10); # Peter Luschny, Jan 26 2017
MATHEMATICA
T[n_] := ((-n-x+1)^n+(x-1)(1-n)^n-(-n)^n x) (-1)^n/((x-1) x); Table[CoefficientList[T[n], x], {n, 2, 10}] // Flatten (* Peter Luschny, Jan 26 2017 *)
PROG
(Sage)
def fermat(n):
q = polygen(ZZ, 'q')
return sum(n**j * binomial(n, j) * (-1)**(i + n + j) *
binomial(n - 2 - j + 1, i + 1) * q**i
for j in range(n - 1)
for i in range(n - 1 - j))
(Sage) # Alternatively:
def A281620_row(n):
x = polygen(ZZ, 'x')
p = (((-n-x+1)^n + (x-1)*(1-n)^n - (-n)^n*x)*(-1)^n)//((x-1)*x)
return p.list()
for n in (2..10): print(A281620_row(n)) # Peter Luschny, Jan 26 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
F. Chapoton, Jan 25 2017
STATUS
approved