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.

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.

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: A128311 A334076 A132884 * A210473 A185951 A188832

Adjacent sequences: A319231 A319232 A319233 * A319235 A319236 A319237

Keyword

sign,tabl

Author

Peter Luschny, Sep 14 2018

Status

approved