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%I #8 Dec 27 2019 16:41:28
%S 1,2,3,14,36,27,590,3024,6156,5832,2187,1044302,10704960,49225968,
%T 132339744,227246796,255091680,182815704,76527504,14348907,
%U 3271700001614,67075266827520,652229166810816,3990988066439808,17193623473530864,55281675697126272
%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) = 3 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, 14, 590, 1044302, 3271700001614, ...) 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 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, 3;
%e 14, 36, 27;
%e 590, 3024, 6156, 5832, 2187;
%e 1044302, 10704960, 49225968, 132339744, 227246796, 255091680, 182815704, 76527504, 14348907.
%e Rows 0..4, the polynomials u(n,x):
%e 1;
%e 2 + 3 x^2;
%e 14 + 36 x^2 + 27 x^4;
%e 590 + 3024 x^2 + 6156 x^4 + 5832 x^6 + 2187 x^8;
%e 1044302 + 10704960 x^2 + 49225968 x^4 + 132339744 x^6 + 227246796 x^8 + 255091680 x^10 + 182815704 x^12 + 76527504
%e x^14 + 14348907 x^16.
%t f[x_] := 3 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}]] (* A329442 polynomials u(n,x) *)
%t Table[CoefficientList[u[n, Sqrt[x], x], {n, 0, 5}] (* A329442 array *)
%Y Cf. A329429, A329430, A329431, A329432, A329441.
%K nonn,tabf
%O 0,2
%A _Clark Kimberling_, Dec 07 2019
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