A287479
Expansion of g.f. (x + x^2)/(1 + 3*x^2).
Original entry on oeis.org
0, 1, 1, -3, -3, 9, 9, -27, -27, 81, 81, -243, -243, 729, 729, -2187, -2187, 6561, 6561, -19683, -19683, 59049, 59049, -177147, -177147, 531441, 531441, -1594323, -1594323, 4782969, 4782969, -14348907, -14348907, 43046721, 43046721, -129140163, -129140163, 387420489
Offset: 0
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Join[{0}, LinearRecurrence[{0, -3}, {1, 1}, 40]]
(* or, computation from b = A157241 : *)
b[n_] := (Switch[Mod[n, 3], 0, (-1)^((n + 3)/3), 1, (-1)^((n + 5)/3), 2, (-1)^((n + 4)/3)*2]*2^n + 1)/3; tb = Table[b[n], {n, 0, 40}]; Table[ Differences[tb, n], {n, 0, 40}][[All, 1]]
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concat([0], Vec((x + x^2)/(1 + 3*x^2) + O(x^40))) \\ Felix Fröhlich, Oct 23 2018
A344914
T(n, k) = 2^(3*k)*(n - 3*k)!, for n >= 0 and 0 <= k <= floor(n/3). Triangle read by rows.
Original entry on oeis.org
1, 1, 2, 6, 8, 24, 8, 120, 16, 720, 48, 64, 5040, 192, 64, 40320, 960, 128, 362880, 5760, 384, 512, 3628800, 40320, 1536, 512, 39916800, 322560, 7680, 1024, 479001600, 2903040, 46080, 3072, 4096, 6227020800, 29030400, 322560, 12288, 4096
Offset: 0
[ 0] 1;
[ 1] 1;
[ 2] 2;
[ 3] 6, 8;
[ 4] 24, 8;
[ 5] 120, 16;
[ 6] 720, 48, 64;
[ 7] 5040, 192, 64;
[ 8] 40320, 960, 128;
[ 9] 362880, 5760, 384, 512;
[10] 3628800, 40320, 1536, 512;
[11] 39916800, 322560, 7680, 1024;
[12] 479001600, 2903040, 46080, 3072, 4096;
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T := (n, k) -> 2^(3*k)*(n-3*k)!: seq(seq(T(n,k), k = 0..n/3), n = 0..13);
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Table[2^(3k) (n-3k)!,{n,0,20},{k,0,Floor[n/3]}]//Flatten (* Harvey P. Dale, Feb 13 2022 *)
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.
Original entry on oeis.org
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
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]
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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] ** P[x, n - 1];
Table[CoefficientList[P[x, n][[1]], x], {n, 0, 10}] // Flatten
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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)*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
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