OFFSET
0,8
COMMENTS
Related to conjecture from entry A047211.
LINKS
Gheorghe Coserea, Rows n = 0..200, flattened
FORMULA
P(n) = det(t*I - M), where M(i,j) = [(i+j>n) or (i+j)=n-1], 1 <= i,j <= n.
P(n) = (2*t + 3*(-1)^n)*P(n-1) - (t^2 - 4)*P(n-2) - (2*t + 3*(-1)^n)*P(n-3) - P(n-4).
G.f.: A(x;t) = Sum_{n>=0} P(n)*x^n = (t*x^8 + (-t^2 + t - 1)*x^7 + (-t^3 + t^2 + 2*t + 1)*x^6 + (t^4 - 2*t^3 + t^2 + 2*t)*x^5 - t^2*x^4 + (-t^3 - t^2 + 2*t)*x^3 + (-t^2 - t)*x^2 + (t - 1)*x + 1)/(x^8 + (-2*t^2 + 1)*x^6 + t^4*x^4 + (-2*t^2 + 1)*x^2 + 1).
EXAMPLE
P(0) = 1;
P(1) = t - 1;
P(2) = t^2 - t - 1;
P(3) = t^3 - 3*t^2 + t + 1;
P(4) = t^4 - 2*t^3 - 4*t^2 + t + 1;
...
Triangle starts:
n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[0] 1
[1] 1, -1;
[2] 1, -1, -1;
[3] 1, -3, 1, 1;
[4] 1, -2, -4, 1, 1;
[5] 1, -4, -1, 6, 1, -1;
[6] 1, -3, -8, 3, 9, 1, -1;
[7] 1, -5, -4, 15, 5, -11, -1, 1;
[8] 1, -4, -13, 8, 27, -3, -14, 1, 1;
[9] 1, -6, -8, 29, 15, -42, -6, 18, -1, -1;
[10] 1, -5, -19, 17, 60, -19, -63, 9, 21, -1, -1;
[11] 1, -7, -13, 49, 35, -110, -29, 93, 6, -25, -1, 1;
[12] 1, -6, -26, 31, 114, -58, -189, 45, 29, -10, -30, -1, 1;
...
For n=7 the n X n matrix M (dots for zeros):
[. . . . 1 . 1]
[. . . 1 . 1 1]
[. . 1 . 1 1 1]
[. 1 . 1 1 1 1]
[1 . 1 1 1 1 1]
[. 1 1 1 1 1 1]
[1 1 1 1 1 1 1]
has characteristic polynomial P(7) = det(t*I-M) = t^7 - 5*t^6 - 4*t^5 + 15*t^4 + 5*t^3 - 11*t^2 - t + 1 (which is irreducible over Q: an elementary check shows that P(7)(25) = 4849680601 is a prime and 25 >= 17 = 2 + max(abs([1,-5,-4,15,5,-11,-1,1]))).
PROG
(PARI)
P(n, t='t) = charpoly(matrix(n, n, i, j, (i+j > n) || (i+j)==n-1), t);
seq(N, t='t) = {
my(a=vector(N)); for (n=1, 4, a[n] = subst(P(n, 't), 't, t));
for (n=5, N,
a[n] += (2*t + 3*(-1)^(n%2))*a[n-1] - (t^2-4)*a[n-2];
a[n] += -(2*t + 3*(-1)^(n%2))*a[n-3] - a[n-4]);
a;
};
concat(1, concat(apply(p->Vec(p), seq(12))))
\\ test: N=100; vector(N, n, P(n)) == seq(N)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Gheorghe Coserea, May 30 2018
STATUS
approved