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proposed

allocated for Clark KimberlingIrregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

1, 2, 1, 6, 4, 1, 38, 48, 28, 8, 1, 1446, 3648, 4432, 3296, 1628, 544, 120, 16, 1, 2090918, 10550016, 26125248, 41867904, 48398416, 42666880, 29610272, 16475584, 7419740, 2711424, 800992, 189248, 35064, 4928, 496, 32, 1, 4371938082726, 44118436709376

0,2

Let f(x) = x^2 + 2, 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)) = (1,2,6,38,1446, ... ) 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. p(n,0)) = A072191(n) for n >= 1.

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

Rows 0..4:

1;

2, 1;

6, 4, 1;

38, 48, 28, 8, 1;

1446, 3648, 4432, 3296, 1628, 544, 120, 16, 1.

Rows 0..4, the polynomials u(n,x):

1;

2 + x^2;

6 + 4 x^2 + x^4;

38 + 48 x^2 + 28 x^4 + 8 x^6 + x^8;

1446 + 3648 x^2 + 4432 x^4 + 3296 x^6 + 1628 x^8 + 544 x^10 + 120 x^12 + 16 x^14 + x^16.

f[x_] := x^2 + 2; 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}]] (* A329431 polynomials u(n, x) *)

Table[CoefficientList[u[n, Sqrt[x], x], {n, 0, 5}] (* A329431 array *)

Cf. A329429, A329430, A329432, A329433.

allocated

nonn,tabf

Clark Kimberling, Nov 23 2019

approved

editing

allocated for Clark Kimberling

allocated

approved