A329442 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

1, 2, 3, 14, 36, 27, 590, 3024, 6156, 5832, 2187, 1044302, 10704960, 49225968, 132339744, 227246796, 255091680, 182815704, 76527504, 14348907, 3271700001614, 67075266827520, 652229166810816, 3990988066439808, 17193623473530864, 55281675697126272

Offset: 0

Author

Clark Kimberling, Dec 07 2019

Keywords

• nonn
• tabf

Comments

Let f(x) = 3 x^2 + 2, 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, 2, 14, 590, 1044302, 3271700001614, ...) 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..4:
  1;
  2, 3;
  14, 36, 27;
  590, 3024, 6156, 5832, 2187;
  1044302, 10704960, 49225968, 132339744, 227246796, 255091680, 182815704, 76527504, 14348907.
Rows 0..4, the polynomials u(n,x):
  1;
  2 + 3 x^2;
  14 + 36 x^2 + 27 x^4;
  590 + 3024 x^2 + 6156 x^4 + 5832 x^6 + 2187 x^8;
  1044302 + 10704960 x^2 + 49225968 x^4 + 132339744 x^6 + 227246796 x^8 + 255091680 x^10 + 182815704 x^12 + 76527504
  x^14 + 14348907 x^16.
		

References

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

Crossrefs

Cf. A329429, A329430, A329431, A329432, A329441.

Programs

Mathematica

f[x_] := 3 x^2 + 2;  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}]] (* A329442 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329442 array *)