cp's OEIS Frontend

A329429 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

%I A329429 #10 Dec 27 2019 16:39:50
%S A329429 1,1,1,2,2,1,5,8,8,4,1,26,80,144,168,138,80,32,8,1,677,4160,13888,
%T A329429 31776,54792,74624,82432,74944,56472,35296,18208,7664,2580,672,128,16,
%U A329429 1,458330,5632640,36109952,158572864,531441232,1439520512,3264101376,6342205824
%N A329429 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.
%C A329429 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 A329429 L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.
%F A329429 p(n,0)  = (1, 1, 2, 5, 26, 677, 458330, ...)
%F A329429 p(n,1)  =    (1, 2, 5, 26, 677, 458330, ...)
%F A329429 p(n,2)  =       (2, 5, 26, 677, 458330, ...)
%F A329429 p(n,5)  =          (5, 26, 677, 458330, ...)
%F A329429 p(n,26) =             (26, 677, 458330, ...), etc.;
%F A329429 that is, p(n,p(k,0)) = p(n+k-2,0); there are similar identities for other sequences p(n,h).
%e A329429 Rows 0..4:
%e A329429    1;
%e A329429    1,  1;
%e A329429    2,  2,   1;
%e A329429    5,  8,   8,   4,   1;
%e A329429   26, 80, 144, 168, 138, 80, 32, 8, 1.
%e A329429 Rows 0..4, the polynomials u(n,x):
%e A329429   1,
%e A329429   1 + x^2,
%e A329429   2 + 2 x^2 + x^4,
%e A329429   5 + 8 x^2 + 8 x^4 + 4 x^6 + x^8,
%e A329429   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 A329429 f[x_] := x^2 + 1;  u[0, x_] := 1;
%t A329429 u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
%t A329429 Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329429 polynomials u(n,x) *)
%t A329429 Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 7}]  (* A329429 array *)
%Y A329429 Cf. A003095, A008318, A329430, A329431, A329432, A329433.
%K A329429 nonn,tabf
%O A329429 0,4
%A A329429 _Clark Kimberling_, Nov 13 2019