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 A181738 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. 2
 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 (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 Peter Luschny, Rows 0..45, flattened Wikipedia, Quaternion EXAMPLE 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}. MATHEMATICA 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 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, 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 CROSSREFS Cf. T(n,0) = A138230, A213421 (row sums). Sequence in context: A246745 A111540 A096440 * A121350 A198569 A135080 Adjacent sequences:  A181735 A181736 A181737 * A181739 A181740 A181741 KEYWORD tabl,sign AUTHOR EXTENSIONS Edited by Peter Luschny, Sep 14 2018 STATUS approved

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Last modified October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)