A329431 - OEIS

%I #8 Dec 18 2024 21:01:04

%S 1,2,1,6,4,1,38,48,28,8,1,1446,3648,4432,3296,1628,544,120,16,1,

%T 2090918,10550016,26125248,41867904,48398416,42666880,29610272,

%U 16475584,7419740,2711424,800992,189248,35064,4928,496,32,1,4371938082726,44118436709376

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

%C 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)).

%C 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.

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

%e Rows 0..4:

%e   1;

%e   2, 1;

%e   6, 4, 1;

%e   38, 48, 28, 8, 1;

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

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

%e   1;

%e   2 + x^2;

%e   6 + 4 x^2 + x^4;

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

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

%t f[x_] := x^2 + 2;  u[0, x_] := 1;

%t u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]

%t Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329431 polynomials u(n,x) *)

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

%Y Cf. A329429, A329430, A329432, A329433.

%K nonn,tabf

%O 0,2

%A _Clark Kimberling_, Nov 23 2019