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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

Table of n, a(n) for n=0..65.

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.<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: 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.)