A329431 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

1, 2, 1, 6, 4, 1, 38, 48, 28, 8, 1, 1446, 3648, 4432, 3296, 1628, 544, 120, 16, 1, 2090918, 10550016, 26125248, 41867904, 48398416, 42666880, 29610272, 16475584, 7419740, 2711424, 800992, 189248, 35064, 4928, 496, 32, 1, 4371938082726, 44118436709376

Offset

0,2

Comments

Let f(x) = 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)) = (1,2,6,38,1446, ... ) 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. p(n,0) = A072191(n) for n >= 1.

References

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

Links

Table of n, a(n) for n=0..38.

Example

Rows 0..4:
  1;
  2, 1;
  6, 4, 1;
  38, 48, 28, 8, 1;
  1446, 3648, 4432, 3296, 1628, 544, 120, 16, 1.
Rows 0..4, the polynomials u(n,x):
  1;
  2 + x^2;
  6 + 4 x^2 + x^4;
  38 + 48 x^2 + 28 x^4 + 8 x^6 + x^8;
  1446 + 3648 x^2 + 4432 x^4 + 3296 x^6 + 1628 x^8 + 544 x^10 + 120 x^12 + 16 x^14 + x^16.

Mathematica

f[x_] := 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}]] (* A329431 polynomials u(n, x) *)
Table[CoefficientList[u[n, Sqrt[x], x], {n, 0, 5}]  (* A329431 array *)

Crossrefs

Cf. A329429, A329430, A329432, A329433.

Sequence in context: A135885 A162312 A141715 * A328923 A263271 A098697

Adjacent sequences: A329428 A329429 A329430 * A329432 A329433 A329434

Keyword

nonn,tabf

Author

Clark Kimberling, Nov 23 2019

Status

approved