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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
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.
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.<x> = QQ[]
K = R.fraction_field()
H.<i, j, k> = 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: A128311 A334076 A132884 * A210473 A185951 A188832
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Sep 14 2018
STATUS
approved