A329442 -id:A329442 - cp's OEIS frontend

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

Clark Kimberling, Dec 07 2019

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.

			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.

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

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

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 *)

cp's OEIS Frontend

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

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