A329430 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

1, 1, 1, 2, 3, 3, 1, 9, 36, 90, 147, 171, 144, 87, 36, 9, 1, 730, 8748, 56862, 257337, 895941, 2528172, 5967108, 12025098, 20984508, 32024268, 43036029, 51168267, 53983503, 50609772, 42164064, 31176036, 20403009, 11768247, 5946156, 2610171, 984420, 314262, 83214, 17766, 2934, 351, 27, 1

Offset: 0

Author

Clark Kimberling, Nov 23 2019

Keywords

• nonn
• tabf

Comments

Let f(x) = x^3 + 1, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)).

Then the sequence (p(n,0)) = (1,1,2,9,730, ... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

Examples

			Rows 0..3:
  1;
  1, 1;
  2, 3, 3, 1;
  9, 36, 90, 147, 171, 144, 87, 36, 9, 1.
Rows 0..3, the polynomials u(n,x):
  1;
  1 + x^3;
  2 + 3 x^3 + 3 x^6 + x^9;
  9 + 36 x^3 + 90 x^6 + 147 x^9 + 171 x^12 + 144 x^15 + 87 x^18 + 36 x^21 + 9 x^24 + x^27.
		

References

• L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Crossrefs

Cf. A329429, A329431, A329432, A329433.

Programs

Mathematica

f[x_] := x^3 + 1;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329430 polynomials u(n,x) *)
Table[CoefficientList[u[n, x^(1/3)], x], {n, 0, 5}]  (* A329430 array *)