%I #29 May 04 2019 00:43:27
%S 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,
%T 1,-5,-4,15,5,-11,-1,1,1,-4,-13,8,27,-3,-14,1,1,1,-6,-8,29,15,-42,-6,
%U 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
%N 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.
%C Related to conjecture from entry A047211.
%H Gheorghe Coserea, <a href="/A305319/b305319.txt">Rows n = 0..200, flattened</a>
%F P(n) = det(t*I - M), where M(i,j) = [(i+j>n) or (i+j)=n-1], 1 <= i,j <= n.
%F 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).
%F 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).
%e P(0) = 1;
%e P(1) = t - 1;
%e P(2) = t^2 - t - 1;
%e P(3) = t^3 - 3*t^2 + t + 1;
%e P(4) = t^4 - 2*t^3 - 4*t^2 + t + 1;
%e ...
%e Triangle starts:
%e n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
%e [0] 1
%e [1] 1, -1;
%e [2] 1, -1, -1;
%e [3] 1, -3, 1, 1;
%e [4] 1, -2, -4, 1, 1;
%e [5] 1, -4, -1, 6, 1, -1;
%e [6] 1, -3, -8, 3, 9, 1, -1;
%e [7] 1, -5, -4, 15, 5, -11, -1, 1;
%e [8] 1, -4, -13, 8, 27, -3, -14, 1, 1;
%e [9] 1, -6, -8, 29, 15, -42, -6, 18, -1, -1;
%e [10 1, -5, -19, 17, 60, -19, -63, 9, 21, -1, -1;
%e [11] 1, -7, -13, 49, 35, -110, -29, 93, 6, -25, -1, 1;
%e [12] 1, -6, -26, 31, 114, -58, -189, 45, 29, -10, -30, -1, 1;
%e ...
%e For n=7 the n X n matrix M (dots for zeros):
%e [. . . . 1 . 1]
%e [. . . 1 . 1 1]
%e [. . 1 . 1 1 1]
%e [. 1 . 1 1 1 1]
%e [1 . 1 1 1 1 1]
%e [. 1 1 1 1 1 1]
%e [1 1 1 1 1 1 1]
%e 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]))).
%o (PARI)
%o P(n, t='t) = charpoly(matrix(n, n, i, j, (i+j > n) || (i+j)==n-1), t);
%o seq(N, t='t) = {
%o my(a=vector(N)); for (n=1, 4, a[n] = subst(P(n,'t), 't, t));
%o for (n=5, N,
%o a[n] += (2*t + 3*(-1)^(n%2))*a[n-1] - (t^2-4)*a[n-2];
%o a[n] += -(2*t + 3*(-1)^(n%2))*a[n-3] - a[n-4]);
%o a;
%o };
%o concat(1, concat(apply(p->Vec(p), seq(12))))
%o \\ test: N=100; vector(N, n, P(n)) == seq(N)
%Y Cf. A047211.
%K sign,tabl
%O 0,8
%A _Gheorghe Coserea_, May 30 2018
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