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%I #14 Sep 15 2018 03:47:40
%S 1,0,1,-3,0,1,0,-9,0,1,9,0,-18,0,1,0,45,0,-30,0,1,-27,0,135,0,-45,0,1,
%T 0,-189,0,315,0,-63,0,1,81,0,-756,0,630,0,-84,0,1,0,729,0,-2268,0,
%U 1134,0,-108,0,1,-243,0,3645,0,-5670,0,1890,0,-135,0,1
%N T(n, k) is the coefficient of x^k of the polynomial p(n) which is defined as the scalar part of P(n) = Q(x, 1, 1, 1) * P(n-1) for n > 0 and P(0) = Q(1, 0, 0, 0) where Q(a, b, c, d) is a quaternion, triangle read by rows.
%C The symbol '*' in the name refers to the noncommutative multiplication in Hamilton's division algebra. Traditionally Q(a, b, c, d) is written a + b*i + c*j + d*k.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Quaternion">Quaternion</a>
%e The list of polynomials starts 1, x, x^2 - 3, x^3 - 9*x, x^4 - 18*x^2 + 9, ... and the list of coefficients of the polynomials starts:
%e [0] [ 1]
%e [1] [ 0, 1]
%e [2] [ -3, 0, 1]
%e [3] [ 0, -9, 0, 1]
%e [4] [ 9, 0, -18, 0, 1]
%e [5] [ 0, 45, 0, -30, 0, 1]
%e [6] [-27, 0, 135, 0, -45, 0, 1]
%e [7] [ 0, -189, 0, 315, 0, -63, 0, 1]
%e [8] [ 81, 0, -756, 0, 630, 0, -84, 0, 1]
%e [9] [ 0, 729, 0, -2268, 0, 1134, 0, -108, 0, 1]
%t Needs["Quaternions`"]
%t P[x_, 0 ] := Quaternion[1, 0, 0, 0];
%t P[x_, n_] := P[x, n] = Quaternion[x, 1, 1, 1] ** P[x, n - 1];
%t Table[CoefficientList[P[x, n][[1]], x], {n, 0, 10}] // Flatten
%o (Sage)
%o R.<x> = QQ[]
%o K = R.fraction_field()
%o H.<i, j, k> = QuaternionAlgebra(K, -1, -1)
%o def Q(a, b, c, d): return H(a + b*i + c*j + d*k)
%o @cached_function
%o def P(n):
%o return Q(x, 1, 1, 1)*P(n-1) if n > 0 else Q(1, 0, 0, 0)
%o def p(n): return P(n)[0].numerator().list()
%o flatten([p(n) for n in (0..10)]) # Kudos to William Stein
%Y Inspired by the sister sequence A181738 of _Roger L. Bagula_.
%Y Cf. A254006 (T(n,0) up to sign), A138230 (row sums).
%K sign,tabl
%O 0,4
%A _Peter Luschny_, Sep 14 2018
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