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A329442
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Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.
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2
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1, 2, 3, 14, 36, 27, 590, 3024, 6156, 5832, 2187, 1044302, 10704960, 49225968, 132339744, 227246796, 255091680, 182815704, 76527504, 14348907, 3271700001614, 67075266827520, 652229166810816, 3990988066439808, 17193623473530864, 55281675697126272
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OFFSET
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0,2
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COMMENTS
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Let f(x) = 3 x^2 + 2, 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, 2, 14, 590, 1044302, 3271700001614, ...) 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.
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REFERENCES
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L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.
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LINKS
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EXAMPLE
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Rows 0..4:
1;
2, 3;
14, 36, 27;
590, 3024, 6156, 5832, 2187;
1044302, 10704960, 49225968, 132339744, 227246796, 255091680, 182815704, 76527504, 14348907.
Rows 0..4, the polynomials u(n,x):
1;
2 + 3 x^2;
14 + 36 x^2 + 27 x^4;
590 + 3024 x^2 + 6156 x^4 + 5832 x^6 + 2187 x^8;
1044302 + 10704960 x^2 + 49225968 x^4 + 132339744 x^6 + 227246796 x^8 + 255091680 x^10 + 182815704 x^12 + 76527504
x^14 + 14348907 x^16.
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MATHEMATICA
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f[x_] := 3 x^2 + 2; 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}]] (* A329442 polynomials u(n, x) *)
Table[CoefficientList[u[n, Sqrt[x], x], {n, 0, 5}] (* A329442 array *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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