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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. 0
 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, 0, -189, 0, 315, 0, -63, 0, 1, 81, 0, -756, 0, 630, 0, -84, 0, 1, 0, 729, 0, -2268, 0, 1134, 0, -108, 0, 1, -243, 0, 3645, 0, -5670, 0, 1890, 0, -135, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. LINKS Wikipedia, Quaternion EXAMPLE 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: [0] [  1] [1] [  0,    1] [2] [ -3,    0,    1] [3] [  0,   -9,    0,     1] [4] [  9,    0,  -18,     0,   1] [5] [  0,   45,    0,   -30,   0,    1] [6] [-27,    0,  135,     0, -45,    0,   1] [7] [  0, -189,    0,   315,   0,  -63,   0,    1] [8] [ 81,    0, -756,     0, 630,    0, -84,    0, 1] [9] [  0,  729,    0, -2268,   0, 1134,   0, -108, 0, 1] MATHEMATICA Needs["Quaternions`"] P[x_, 0 ] := Quaternion[1, 0, 0, 0]; P[x_, n_] := P[x, n] = Quaternion[x, 1, 1, 1] ** P[x, n - 1]; Table[CoefficientList[P[x, n][[1]], x], {n, 0, 10}] // Flatten PROG (Sage) R. = QQ[] K = R.fraction_field() H. = QuaternionAlgebra(K, -1, -1) def Q(a, b, c, d): return H(a + b*i + c*j + d*k) @cached_function def P(n):     return Q(x, 1, 1, 1)*P(n-1) if n > 0 else Q(1, 0, 0, 0) def p(n): return P(n)[0].numerator().list() flatten([p(n) for n in (0..10)]) # Kudos to William Stein CROSSREFS Inspired by the sister sequence A181738 of Roger L. Bagula. Cf. A254006 (T(n,0) up to sign), A138230 (row sums). Sequence in context: A238123 A128311 A132884 * A210473 A185951 A188832 Adjacent sequences:  A319231 A319232 A319233 * A319235 A319236 A319237 KEYWORD sign,tabl AUTHOR Peter Luschny, Sep 14 2018 STATUS approved

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Last modified April 8 06:35 EDT 2020. Contains 333312 sequences. (Running on oeis4.)