A329432 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

1, 1, 2, 3, 8, 8, 19, 96, 224, 256, 128, 723, 7296, 35456, 105472, 208384, 278528, 245760, 131072, 32768, 1045459, 21100032, 209001984, 1339772928, 6194997248, 21845442560, 60641837056, 134967984128, 243130040320, 355391766528, 419950493696, 396881821696

Offset

0,3

Comments

Let f(x) = 2 x^2 + 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, 3, 19, 723, 1045459, ... ) 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.

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..31.

Example

Rows 0..4:
  1;
  1, 2;
  3, 8, 8;
  19, 96, 224, 256, 128;
  723, 7296, 35456, 105472, 208384, 278528, 245760, 131072, 32768.
Rows 0..4, the polynomials u(n,x):
  1,
  1 + 2 x^2
  3 + 8 x^2 + 8 x^4
  19 + 96 x^2 + 224 x^4 + 256 x^6 + 128 x^8
  723 + 7296 x^2 + 35456 x^4 + 105472 x^6 + 208384 x^8 + 278528 x^10 + 245760 x^12 + 131072 x^14 + 32768 x^16.

Mathematica

f[x_] := 2 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}]] (* A329432 polynomials u(n, x) *)
Table[CoefficientList[u[n, Sqrt[x], x], {n, 0, 5}]  (* A329432 array *)

Crossrefs

Cf. A329429, A329430, A329431, A329433.

Sequence in context: A237643 A225474 A368235 * A100805 A068800 A291838

Adjacent sequences: A329429 A329430 A329431 * A329433 A329434 A329435

Keyword

nonn,tabf

Author

Clark Kimberling, Nov 23 2019

Status

approved