A367301 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2*x - x^2.

1, 3, 3, 10, 16, 8, 33, 75, 63, 21, 109, 320, 380, 220, 55, 360, 1296, 1980, 1620, 720, 144, 1189, 5070, 9459, 9940, 6255, 2262, 377, 3927, 19353, 42615, 54561, 44085, 22635, 6909, 987, 12970, 72532, 184034, 277480, 272854, 179972, 78230, 20672, 2584

Offset: 1

Author

Clark Kimberling, Dec 23 2023

Keywords

• nonn
• tabl

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
     1
     3      3
    10     16      8
    33     75     63     21
   109    320    380    220     55
   360   1296   1980   1620    720    144
  1189   5070   9459   9940   6255   2262   377
  3927  19353  42615  54561  44085  22635  6909  987
Row 4 represents the polynomial p(4,x) = 33 + 75*x + 63*x^2 + 21*x^3, so (T(4,k)) = (33,75,63,21), k=0..3.
		

Links

• Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Crossrefs

Cf. A006190 (column 1); A001906 (p(n,n-1)); A154244 (row sums, p(n,1)); A077957 (alternating row sums, p(n,-1)); A190984 (p(n,2)); A006190 (signed, p(n,-2)); A154244 (p(n,-3)); A190984 (p(n,-4)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300.

Programs

Mathematica

p[1, x_] := 1; p[2, x_] := 3 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 3 + 3*x, u = p(2,x), and v = 1 - 2*x - x^2.

p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(13 + 10*x + 5*x^2)), b = (1/2) (3*x + 3 + 1/k), c = (1/2) (3*x + 3 - 1/k).