A305319 Triangle T(n,k) read by rows: coefficients in order of decreasing exponents of characteristic polynomial P_n(t) of the matrix M(i,j) = [(i+j>n) or (i+j)=n-1], 1 <= i,j <= n.

1, 1, -1, 1, -1, -1, 1, -3, 1, 1, 1, -2, -4, 1, 1, 1, -4, -1, 6, 1, -1, 1, -3, -8, 3, 9, 1, -1, 1, -5, -4, 15, 5, -11, -1, 1, 1, -4, -13, 8, 27, -3, -14, 1, 1, 1, -6, -8, 29, 15, -42, -6, 18, -1, -1, 1, -5, -19, 17, 60, -19, -63, 9, 21, -1, -1, 1, -7, -13, 49, 35, -110, -29, 93, 6, -25, -1, 1, 1, -6, -26, 31, 114, -58, -189, 45, 129, -10, -30, -1, 1

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(tI-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

Cf. A047211.

Sequence in context: A136406 A242222 A247198 * A026568 A138361 A030408

Adjacent sequences: A305316 A305317 A305318 * A305320 A305321 A305322

Keyword

sign,tabl

Author

Gheorghe Coserea, May 30 2018

Status

approved