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A329430 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

%I A329430 #7 Jun 19 2025 23:51:39
%S A329430 1,1,1,2,3,3,1,9,36,90,147,171,144,87,36,9,1,730,8748,56862,257337,
%T A329430 895941,2528172,5967108,12025098,20984508,32024268,43036029,51168267,
%U A329430 53983503,50609772,42164064,31176036,20403009,11768247,5946156,2610171,984420,314262,83214,17766,2934,351,27,1
%N A329430 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.
%C A329430 Let f(x) = x^3 + 1, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)).
%C A329430 Then the sequence (p(n,0)) = (1,1,2,9,730, ... ) 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 A329430 L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.
%e A329430 Rows 0..3:
%e A329430   1;
%e A329430   1, 1;
%e A329430   2, 3, 3, 1;
%e A329430   9, 36, 90, 147, 171, 144, 87, 36, 9, 1.
%e A329430 Rows 0..3, the polynomials u(n,x):
%e A329430   1;
%e A329430   1 + x^3;
%e A329430   2 + 3 x^3 + 3 x^6 + x^9;
%e A329430   9 + 36 x^3 + 90 x^6 + 147 x^9 + 171 x^12 + 144 x^15 + 87 x^18 + 36 x^21 + 9 x^24 + x^27.
%t A329430 f[x_] := x^3 + 1;  u[0, x_] := 1;
%t A329430 u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
%t A329430 Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329430 polynomials u(n,x) *)
%t A329430 Table[CoefficientList[u[n, x^(1/3)], x], {n, 0, 5}]  (* A329430 array *)
%Y A329430 Cf. A329429, A329431, A329432, A329433.
%K A329430 nonn,tabf
%O A329430 0,4
%A A329430 _Clark Kimberling_, Nov 23 2019