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Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.
8

%I #10 Dec 27 2019 16:39:50

%S 1,1,1,2,2,1,5,8,8,4,1,26,80,144,168,138,80,32,8,1,677,4160,13888,

%T 31776,54792,74624,82432,74944,56472,35296,18208,7664,2580,672,128,16,

%U 1,458330,5632640,36109952,158572864,531441232,1439520512,3264101376,6342205824

%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) = x^2 + 1, u(0,x) = 1, u(n,x) = f(u(n-1,x)), and p(n,x) = u(n,sqrt(x)). Except for the first term, the sequence (p(n,0)) = (1, 1, 5, 26, 677, ...) is found in A003095 and A008318. This 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.

%F p(n,0) = (1, 1, 2, 5, 26, 677, 458330, ...)

%F p(n,1) = (1, 2, 5, 26, 677, 458330, ...)

%F p(n,2) = (2, 5, 26, 677, 458330, ...)

%F p(n,5) = (5, 26, 677, 458330, ...)

%F p(n,26) = (26, 677, 458330, ...), etc.;

%F that is, p(n,p(k,0)) = p(n+k-2,0); there are similar identities for other sequences p(n,h).

%e Rows 0..4:

%e 1;

%e 1, 1;

%e 2, 2, 1;

%e 5, 8, 8, 4, 1;

%e 26, 80, 144, 168, 138, 80, 32, 8, 1.

%e Rows 0..4, the polynomials u(n,x):

%e 1,

%e 1 + x^2,

%e 2 + 2 x^2 + x^4,

%e 5 + 8 x^2 + 8 x^4 + 4 x^6 + x^8,

%e 26 + 80 x^2 + 144 x^4 + 168 x^6 + 138 x^8 + 80 x^10 + 32 x^12 + 8 x^14 + x^16.

%t f[x_] := x^2 + 1; 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}]] (* A329429 polynomials u(n,x) *)

%t Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 7}] (* A329429 array *)

%Y Cf. A003095, A008318, A329430, A329431, A329432, A329433.

%K nonn,tabf

%O 0,4

%A _Clark Kimberling_, Nov 13 2019