A281620 Triangle read by rows: Poincaré polynomials of orbifold of Fermat hypersurfaces.

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, 102207817, 26710345, 4690249, 561481, 45385, 2377, 73, 1, 2639010709, 701908264, 127951984, 16349374, 1469026, 91216, 3736, 91, 1

Offset: 2

Author

F. Chapoton, Jan 25 2017

Keywords

• nonn
• tabl

Examples

			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;
  ...
		

Links

• So Okada, Homological mirror symmetry of Fermat polynomials, arxiv:0910.2014 [math.AG], 2009-2010.

Crossrefs

Row sums give A007778(n-1), alternating row sums are A281596.

Programs

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 *)

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

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)