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A319234 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.

%I A319234 #14 Sep 15 2018 03:47:40
%S A319234 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 A319234 0,-189,0,315,0,-63,0,1,81,0,-756,0,630,0,-84,0,1,0,729,0,-2268,0,
%U A319234 1134,0,-108,0,1,-243,0,3645,0,-5670,0,1890,0,-135,0,1
%N A319234 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 A319234 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 A319234 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quaternion">Quaternion</a>
%e A319234 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 A319234 [0] [  1]
%e A319234 [1] [  0,    1]
%e A319234 [2] [ -3,    0,    1]
%e A319234 [3] [  0,   -9,    0,     1]
%e A319234 [4] [  9,    0,  -18,     0,   1]
%e A319234 [5] [  0,   45,    0,   -30,   0,    1]
%e A319234 [6] [-27,    0,  135,     0, -45,    0,   1]
%e A319234 [7] [  0, -189,    0,   315,   0,  -63,   0,    1]
%e A319234 [8] [ 81,    0, -756,     0, 630,    0, -84,    0, 1]
%e A319234 [9] [  0,  729,    0, -2268,   0, 1134,   0, -108, 0, 1]
%t A319234 Needs["Quaternions`"]
%t A319234 P[x_, 0 ] := Quaternion[1, 0, 0, 0];
%t A319234 P[x_, n_] := P[x, n] = Quaternion[x, 1, 1, 1] ** P[x, n - 1];
%t A319234 Table[CoefficientList[P[x, n][[1]], x], {n, 0, 10}] // Flatten
%o A319234 (Sage)
%o A319234 R.<x> = QQ[]
%o A319234 K = R.fraction_field()
%o A319234 H.<i, j, k> = QuaternionAlgebra(K, -1, -1)
%o A319234 def Q(a, b, c, d): return H(a + b*i + c*j + d*k)
%o A319234 @cached_function
%o A319234 def P(n):
%o A319234     return Q(x, 1, 1, 1)*P(n-1) if n > 0 else Q(1, 0, 0, 0)
%o A319234 def p(n): return P(n)[0].numerator().list()
%o A319234 flatten([p(n) for n in (0..10)]) # Kudos to William Stein
%Y A319234 Inspired by the sister sequence A181738 of _Roger L. Bagula_.
%Y A319234 Cf. A254006 (T(n,0) up to sign), A138230 (row sums).
%K A319234 sign,tabl
%O A319234 0,4
%A A319234 _Peter Luschny_, Sep 14 2018