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A328647
Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (1+x)/(x^2-3x+1)).
2
1, 1, 4, -2, -1, 11, -12, 3, 1, 29, -44, 24, -4, -1, 76, -145, 110, -40, 5, 1, 199, -456, 435, -220, 60, -6, -1, 521, -1393, 1596, -1015, 385, -84, 7, 1, 1364, -4168, 5572, -4256, 2030, -616, 112, -8, -1, 3571, -12276, 18756, -16716, 9576, -3654, 924, -144
OFFSET
0,3
COMMENTS
The first 201 polynomials are irreducible. Column 1 of the array: A002879 (odd-indexed Lucas numbers). Row sums: A000032 (Lucas numbers). Alternating row sums: essentially 5*A030191.
EXAMPLE
First eight rows:
1, 1;
4, -2, -1;
11, -12, 3, 1;
29, -44, 24, -4, -1;
76, -145, 110, -40, 5, 1;
199, -456, 435, -220, 60, -6, -1;
521, -1393, 1596, -1015, 385, -84, 7, 1;
1364, -4168, 5572, -4256, 2030, -616, 112, -8, -1;
First eight polynomials:
1 + x
4 - 2 x - x^2
11 - 12 x + 3 x^2 + x^3
29 - 44 x + 24 x^2 - 4 x^3 - x^4
76 - 145 x + 110 x^2 - 40 x^3 + 5 x^4 + x^5
199 - 456 x + 435 x^2 - 220 x^3 + 60 x^4 - 6 x^5 - x^6
521 - 1393 x + 1596 x^2 - 1015 x^3 + 385 x^4 - 84 x^5 + 7 x^6 + x^7
1364 - 4168 x + 5572 x^2 - 4256 x^3 + 2030 x^4 - 616 x^5 + 112 x^6 - 8 x^7 - x^8
MATHEMATICA
g[x_, n_] := Numerator[ Factor[D[(1 + x)/(x^2 - 3 x + 1), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x]
Table[h[n], {n, 0, 10}]
Column[%] (* A328647 array *)
CROSSREFS
KEYWORD
tabf,sign
AUTHOR
Clark Kimberling, Nov 01 2019
STATUS
approved