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A281620 Triangle read by rows: Poincaré polynomials of orbifold of Fermat hypersurfaces. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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))

# Alternatively:

def A281620_row(n):

    p = factor(((-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: A051339 A134141 A237111 * A321187 A221367 A110788

Adjacent sequences:  A281617 A281618 A281619 * A281621 A281622 A281623

KEYWORD

nonn,tabl

AUTHOR

F. Chapoton, Jan 25 2017

STATUS

approved

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Last modified February 21 04:18 EST 2019. Contains 320371 sequences. (Running on oeis4.)