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

0,4

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.

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..44.

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).

Example

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.

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 *)

Crossrefs

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

Sequence in context: A297628 A342722 A344528 * A326617 A371727 A326962

Adjacent sequences: A329426 A329427 A329428 * A329430 A329431 A329432

Keyword

nonn,tabf

Author

Clark Kimberling, Nov 13 2019

Status

approved