A329430 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

1, 1, 1, 2, 3, 3, 1, 9, 36, 90, 147, 171, 144, 87, 36, 9, 1, 730, 8748, 56862, 257337, 895941, 2528172, 5967108, 12025098, 20984508, 32024268, 43036029, 51168267, 53983503, 50609772, 42164064, 31176036, 20403009, 11768247, 5946156, 2610171, 984420, 314262

Offset

0,4

Comments

Let f(x) = x^3 + 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)) = (1,1,2,9,730, ... ) 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..38.

Example

Rows 0..3:
  1;
  1, 1;
  2, 3, 3, 1;
  9, 36, 90, 147, 171, 144, 87, 36, 9, 1.
Rows 0..3, the polynomials u(n,x):
  1;
  1 + x^3;
  2 + 3 x^3 + 3 x^6 + x^9;
  9 + 36 x^3 + 90 x^6 + 147 x^9 + 171 x^12 + 144 x^15 + 87 x^18 + 36 x^21 + 9 x^24 + x^27.

Mathematica

f[x_] := x^3 + 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}]] (* A329430 polynomials u(n, x) *)
Table[CoefficientList[u[n, x^(1/3), x], {n, 0, 5}]  (* A329430 array *)

Crossrefs

Cf. A329429, A329431, A329432, A329433.

Sequence in context: A097005 A068008 A123948 * A188886 A131012 A083057

Adjacent sequences: A329427 A329428 A329429 * A329431 A329432 A329433

Keyword

nonn,tabf

Author

Clark Kimberling, Nov 23 2019

Status

approved