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

2,2

Links

Table of n, a(n) for n=2..46.

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

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

Sequence in context: A134141 A350202 A237111 * A321187 A221367 A110788

Adjacent sequences: A281617 A281618 A281619 * A281621 A281622 A281623

Keyword

nonn,tabl

Author

F. Chapoton, Jan 25 2017

Status

approved