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A329433 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

%I A329433 #7 Jun 19 2025 23:51:02
%S A329433 1,3,1,12,6,1,147,144,60,12,1,21612,42336,38376,20808,7350,1728,264,
%T A329433 24,1,467078547,1829931264,3451101120,4148777664,3552268752,
%U A329433 2294085888,1154824416,461895840,148272828,38314944,7942320,1306800,167340,16128,1104,48,1
%N A329433 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.
%C A329433 Let f(x) = x^2 + 3, 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, 3, 12, 147, 21612, 467078547,... ) 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.
%D A329433 L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.
%e A329433 Rows 0..4:
%e A329433   1;
%e A329433   3, 1;
%e A329433   12, 6, 1;
%e A329433   147, 144, 60, 12, 1;
%e A329433   21612, 42336, 38376, 20808, 7350, 1728, 264, 24, 1.
%e A329433 Rows 0..4, the polynomials u(n,x):
%e A329433   1;
%e A329433   3 + x^2;
%e A329433   12 + 6 x^2 + x^4;
%e A329433   147 + 144 x^2 + 60 x^4 + 12 x^6 + x^8;
%e A329433   21612 + 42336 x^2 + 38376 x^4 + 20808 x^6 + 7350 x^8 + 1728 x^10 + 264 x^12 + 24 x^14 + x^16.
%t A329433 f[x_] := x^2 + 3;  u[0, x_] := 1;
%t A329433 u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
%t A329433 Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329433 polynomials u(n,x) *)
%t A329433 Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329433 array *)
%Y A329433 Cf. A329429, A329430, A329431, A329432.
%K A329433 nonn,tabf
%O A329433 0,2
%A A329433 _Clark Kimberling_, Nov 23 2019