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A329431 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

%I A329431 #11 Jun 19 2025 23:51:23
%S A329431 1,2,1,6,4,1,38,48,28,8,1,1446,3648,4432,3296,1628,544,120,16,1,
%T A329431 2090918,10550016,26125248,41867904,48398416,42666880,29610272,
%U A329431 16475584,7419740,2711424,800992,189248,35064,4928,496,32,1,4371938082726,44118436709376
%N A329431 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.
%C A329431 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 A329431 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 A329431 L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.
%e A329431 Rows 0..4:
%e A329431   1;
%e A329431   2, 1;
%e A329431   6, 4, 1;
%e A329431   38, 48, 28, 8, 1;
%e A329431   1446, 3648, 4432, 3296, 1628, 544, 120, 16, 1.
%e A329431 Rows 0..4, the polynomials u(n,x):
%e A329431   1;
%e A329431   2 + x^2;
%e A329431   6 + 4 x^2 + x^4;
%e A329431   38 + 48 x^2 + 28 x^4 + 8 x^6 + x^8;
%e A329431   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 A329431 f[x_] := x^2 + 2;  u[0, x_] := 1;
%t A329431 u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
%t A329431 Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329431 polynomials u(n,x) *)
%t A329431 Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329431 array *)
%Y A329431 Cf. A329429, A329430, A329432, A329433.
%K A329431 nonn,tabf
%O A329431 0,2
%A A329431 _Clark Kimberling_, Nov 23 2019