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A329441 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

%I A329441 #10 Jun 19 2025 23:50:57
%S A329441 1,3,2,21,24,8,885,2016,1824,768,128,1566453,7136640,14585472,
%T A329441 17427456,13300224,6635520,2113536,393216,32768,4907550002421,
%U A329441 44716844551680,193253086462464,525562214510592,1006302608418816,1438003249348608,1586056913289216
%N A329441 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.
%C A329441 Let f(x) = 2 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, 21, 885, 1566453, 4907550002421, 48168094052524714211722485, ... ) 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 A329441 L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.
%e A329441 Rows 0..4:
%e A329441   1;
%e A329441   3, 2;
%e A329441   21, 24, 8;
%e A329441   885, 2016, 1824, 768, 128;
%e A329441   1566453, 7136640, 14585472, 17427456, 13300224, 6635520, 2113536, 393216, 32768.
%e A329441 Rows 0..4, the polynomials u(n,x):
%e A329441   1;
%e A329441   3 + 2 x^2;
%e A329441   21 + 24 x^2 + 8 x^4;
%e A329441   885 + 2016 x^2 + 1824 x^4 + 768 x^6 + 128 x^8;
%e A329441   1566453 + 7136640 x^2 + 14585472 x^4 + 17427456 x^6 + 13300224 x^8 + 6635520 + x^10 + 2113536 x^12 + 393216 x^14 +
%e A329441   32768 x^16.
%t A329441 f[x_] := 2 x^2 + 3;  u[0, x_] := 1;
%t A329441 u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
%t A329441 Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329441 polynomials u(n,x) *)
%t A329441 Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329441 array *)
%Y A329441 Cf. A329429, A329430, A329431, A329432, A329442.
%K A329441 nonn,tabf
%O A329441 0,2
%A A329441 _Clark Kimberling_, Dec 07 2019