A329429 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

1, 1, 1, 2, 2, 1, 5, 8, 8, 4, 1, 26, 80, 144, 168, 138, 80, 32, 8, 1, 677, 4160, 13888, 31776, 54792, 74624, 82432, 74944, 56472, 35296, 18208, 7664, 2580, 672, 128, 16, 1, 458330, 5632640, 36109952, 158572864, 531441232, 1439520512, 3264101376, 6342205824

Offset: 0

Author

Clark Kimberling, Nov 13 2019

Keywords

• nonn
• tabf

Comments

Let f(x) = x^2 + 1, u(0,x) = 1, u(n,x) = f(u(n-1,x)), and p(n,x) = u(n,sqrt(x)). Except for the first term, the sequence (p(n,0)) = (1, 1, 5, 26, 677, ...) is found in A003095 and A008318. This 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;
   1,  1;
   2,  2,   1;
   5,  8,   8,   4,   1;
  26, 80, 144, 168, 138, 80, 32, 8, 1.
Rows 0..4, the polynomials u(n,x):
  1,
  1 + x^2,
  2 + 2 x^2 + x^4,
  5 + 8 x^2 + 8 x^4 + 4 x^6 + x^8,
  26 + 80 x^2 + 144 x^4 + 168 x^6 + 138 x^8 + 80 x^10 + 32 x^12 + 8 x^14 + x^16.
		

References

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

Crossrefs

Cf. A003095, A008318, A329430, A329431, A329432, A329433.

Programs

Mathematica

f[x_] := x^2 + 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}]] (* A329429 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 7}]  (* A329429 array *)

Formula

p(n,0) = (1, 1, 2, 5, 26, 677, 458330, ...)

p(n,1) = (1, 2, 5, 26, 677, 458330, ...)

p(n,2) = (2, 5, 26, 677, 458330, ...)

p(n,5) = (5, 26, 677, 458330, ...)

p(n,26) = (26, 677, 458330, ...), etc.;

that is, p(n,p(k,0)) = p(n+k-2,0); there are similar identities for other sequences p(n,h).