A267611 Irregular triangle array of coefficients in ascending order of modified polynomials that arise in the study of Erdős-Ko-Rado sets of generators of the hyperbolic quadrics Q+(4n + 1, q).

1, 1, 1, -1, 1, 0, 1, 1, 1, -1, 0, -1, 1, -1, 1, 0, 1, 1, -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, 0, 1, 1, 1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -1, 2, -2, 1, -2, 2, -2, 1, -1, 1, -1, 1, 0, 1, 1, -1, 1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 2, -2, 3, -3, 3, -2, 3, -3, 3, -2, 2, -3, 2, -2, 2, -2, 1, -1, 1, -1, 1, 0, 1, 1

Offset

0,27

Comments

The n-th polynomial, as described in the link, is divisible by x^(n*(n-1)).

So they are here divided by this factor, thus removing the n*(n-1) leading zeros from the n-th row.

The initial degree of the n-th polynomial was n*(2*n-1) (A000384), so, finally the length of the n-th row is n^2 + 1 (A002522).

It appears this sequence is identical to A178666 up to the signs.

Links

Table of n, a(n) for n=0..97.

Maarten De Boeck, The second largest Erdős-Ko-Rado sets of generators of the hyperbolic quadrics Q+(4n+1,q), arXiv:1601.03543 [math.CO], 2016. See page 5.

Formula

Pol(n) = (q^(n^2-n)*Prod_{k=1..n} (q^(2*k-1)-1) + 2*Sum_{i=0..n-1} gauss_binomial(2*n,2*i,q)*q^(i^2-i)*Prod_{k=1..i}(q^(2*k-1)-1))/q^(n^2-n), where gauss_binomial(j,k,q) = Prod_{i=1..k} (q^(j+1-i)-1)/(q^i-1)).

Example

The rows with leading zeros are:
[1],
[1, 1],
[0, 0, -1, 1, 0, 1, 1],
[0, 0, 0, 0, 0, 0, 1, -1, 0, -1, 1, -1, 1, 0, 1, 1].
The leading zeros are dropped, so the triangle starts:
[1],
[1, 1],
[-1, 1, 0, 1, 1],
[1, -1, 0, -1, 1, -1, 1, 0, 1, 1],
[-1, 1, 0, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, 0, 1, 1].

Mathematica

gbinom[n_, k_, q_] := Product[(q^(n + 1 - i) - 1)/(q^i - 1), {i, 1, k}];
g[n_] := (q^(n^2 - n) Product[q^(2k-1)-1, {k, 1, n}] + 2 Sum[gbinom[2n, 2i, q] q^(i^2-i) Product[q^(2k-1)-1, {k, 1, i}], {i, 0, n-1}])/q^(n^2-n);
row[n_] := CoefficientList[g[n], q];
Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Sep 27 2018, from PARI *)

Prog

(PARI) gbinom(n, k, q) = prod(i=1, k, (q^(n+1-i)-1)/(q^i-1));
g(n) = (q^(n^2-n)*prod(k=1, n, q^(2*k-1)-1) + 2*sum(i=0, n-1, gbinom(2*n, 2*i, q)*q^(i^2-i)*prod(k=1, i, q^(2*k-1)-1)))/q^(n^2-n);
row(n) = Vecrev(g(n));

Crossrefs

Cf. A000384, A002378, A002522, A178666.

Sequence in context: A181506 A319797 A169987 * A178666 A206706 A302354

Adjacent sequences: A267608 A267609 A267610 * A267612 A267613 A267614

Keyword

sign,tabf

Author

Michel Marcus, Jan 18 2016

Status

approved