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A181738
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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, 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.
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2
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1, 1, 1, -2, 2, 1, -8, -6, 3, 1, -8, -32, -12, 4, 1, 16, -40, -80, -20, 5, 1, 64, 96, -120, -160, -30, 6, 1, 64, 448, 336, -280, -280, -42, 7, 1, -128, 512, 1792, 896, -560, -448, -56, 8, 1, -512, -1152, 2304, 5376, 2016, -1008, -672, -72, 9, 1, -512, -5120, -5760, 7680, 13440, 4032, -1680, -960, -90, 10, 1
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table;
graph;
refs;
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history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The list of polynomials starts 1, 1 + x, -2 + 2*x + x^2, -8 - 6*x + 3*x^2 + x^3, ... and the list of coefficients of the polynomials starts:
{ 1},
{ 1, 1},
{ -2, 2, 1},
{ -8, -6, 3, 1},
{ -8, -32, -12, 4, 1},
{ 16, -40, -80, -20, 5, 1},
{ 64, 96, -120, -160, -30, 6, 1},
{ 64, 448, 336, -280, -280, -42, 7, 1},
{-128, 512, 1792, 896, -560, -448, -56, 8, 1},
{-512, -1152, 2304, 5376, 2016, -1008, -672, -72, 9, 1},
{-512, -5120, -5760, 7680, 13440, 4032, -1680, -960, -90, 10, 1}.
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MATHEMATICA
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Needs["Quaternions`"]
P[x_, 0 ] := Quaternion[1, 0, 0, 0];
P[x_, n_] := P[x, n] = Quaternion[x + 1, 1, 1, 1] ** P[x, n - 1];
Table[CoefficientList[P[x, n][[1]], x], {n, 0, 10}] // Flatten
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PROG
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(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, 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, Peter Luschny, Sep 14 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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