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A329432
Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.
8
1, 1, 2, 3, 8, 8, 19, 96, 224, 256, 128, 723, 7296, 35456, 105472, 208384, 278528, 245760, 131072, 32768, 1045459, 21100032, 209001984, 1339772928, 6194997248, 21845442560, 60641837056, 134967984128, 243130040320, 355391766528, 419950493696, 396881821696
OFFSET
0,3
COMMENTS
Let f(x) = 2 x^2 + 1, 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, 1, 3, 19, 723, 1045459, ... ) 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.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.
EXAMPLE
Rows 0..4:
1;
1, 2;
3, 8, 8;
19, 96, 224, 256, 128;
723, 7296, 35456, 105472, 208384, 278528, 245760, 131072, 32768.
Rows 0..4, the polynomials u(n,x):
1,
1 + 2 x^2
3 + 8 x^2 + 8 x^4
19 + 96 x^2 + 224 x^4 + 256 x^6 + 128 x^8
723 + 7296 x^2 + 35456 x^4 + 105472 x^6 + 208384 x^8 + 278528 x^10 + 245760 x^12 + 131072 x^14 + 32768 x^16.
MATHEMATICA
f[x_] := 2 x^2 + 1; u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329432 polynomials u(n, x) *)
Table[CoefficientList[u[n, Sqrt[x], x], {n, 0, 5}] (* A329432 array *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Nov 23 2019
STATUS
approved