A329441 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

1, 3, 2, 21, 24, 8, 885, 2016, 1824, 768, 128, 1566453, 7136640, 14585472, 17427456, 13300224, 6635520, 2113536, 393216, 32768, 4907550002421, 44716844551680, 193253086462464, 525562214510592, 1006302608418816, 1438003249348608, 1586056913289216

Offset

0,2

Comments

Let f(x) = 2 x^2 + 3, 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, 3, 21, 885, 1566453, 4907550002421, 48168094052524714211722485, ... ) 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..26.

Example

Rows 0..4:
  1;
  3, 2;
  21, 24, 8;
  885, 2016, 1824, 768, 128;
  1566453, 7136640, 14585472, 17427456, 13300224, 6635520, 2113536, 393216, 32768.
Rows 0..4, the polynomials u(n,x):
  1;
  3 + 2 x^2;
  21 + 24 x^2 + 8 x^4;
  885 + 2016 x^2 + 1824 x^4 + 768 x^6 + 128 x^8;
  1566453 + 7136640 x^2 + 14585472 x^4 + 17427456 x^6 + 13300224 x^8 + 6635520 + x^10 + 2113536 x^12 + 393216 x^14 +
  32768 x^16.

Mathematica

f[x_] := 2 x^2 + 3;  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}]] (* A329441 polynomials u(n, x) *)
Table[CoefficientList[u[n, Sqrt[x], x], {n, 0, 5}]  (* A329441 array *)

Crossrefs

Cf. A329429, A329430, A329431, A329432, A329442.

Sequence in context: A350290 A248123 A018872 * A151429 A355290 A151475

Adjacent sequences: A329438 A329439 A329440 * A329442 A329443 A329444

Keyword

nonn,tabf

Author

Clark Kimberling, Dec 07 2019

Status

approved