A326926
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/(1-x+x^2)).
Original entry on oeis.org
1, 1, -2, 0, -3, 3, -1, 0, 6, -4, -1, 5, 0, -10, 5, 0, 6, -15, 0, 15, -6, 1, 0, -21, 35, 0, -21, 7, 1, -8, 0, 56, -70, 0, 28, -8, 0, -9, 36, 0, -126, 126, 0, -36, 9, -1, 0, 45, -120, 0, 252, -210, 0, 45, -10, -1, 11, 0, -165, 330, 0, -462, 330, 0, -55, 11, 0
Offset: 0
First eight rows:
1;
1, -2;
0, -3, 3;
-1, 0, 6, -4;
-1, 5, 0, -10, 5;
0, 6, -15, 0, 15, -6;
1, 0, -21, 35, 0, -21, 7;
1, -8, 0, 56, -70, 0, 28, -8;
First eight polynomials:
1
1 - 2*x
-3*x + 3*x^2 = 3 (-1 + x)*x
-1 + 6*x^2 - 4*x^3 = (-1 + 2*x) (1 + 2*x - 2*x^2)
-1 + 5*x - 10*x^3 + 5*x^4
6*x - 15*x^2 + 15*x^4 - 6*x^5 = -3*x*(-2 + x)*(-1 + x)*(1 + x)*(-1 + 2*x)
1 - 21*x^2 + 35*x^3 - 21*x^5 + 7*x^6
1 - 8*x + 56*x^3 - 70*x^4 + 28*x^6 - 8*x^7 = -(-1 + 2*x)*(-1 - 2*x + 2*x^2)*(-1 + 8*x - 6*x^2 - 4*x^3 + 2*x^4)
-
g[x_, n_] := Numerator[ Factor[D[1/(x^2 - 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}] (* A326926 *)
Column[%]
Table[-1 + Length[FactorList[g[x, n]/n!]], {n, 0, 100}] (* A326933 *)
A328645
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/(1-3x+x^2)).
Original entry on oeis.org
1, 3, -2, 8, -9, 3, 21, -32, 18, -4, 55, -105, 80, -30, 5, 144, -330, 315, -160, 45, -6, 377, -1008, 1155, -735, 280, -63, 7, 987, -3016, 4032, -3080, 1470, -448, 84, -8, 2584, -8883, 13572, -12096, 6930, -2646, 672, -108, 9, 6765, -25840, 44415, -45240
Offset: 0
First eight rows:
1;
3, -2;
8, -9, 3;
21, -32, 18, -4;
55, -105, 80, -30, 5;
144, -330, 315, -160, 45, -6;
377, -1008, 1155, -735, 280, -63, 7;
987, -3016, 4032, -3080, 1470, -448, 84, -8;
First eight polynomials:
1
3 - 2 x
8 - 9 x + 3 x^2
21 - 32 x + 18 x^2 - 4 x^3
= (3 - 2 x) (7 - 6 x + 2 x^2)
55 - 105 x + 80 x^2 - 30 x^3 + 5 x^4
144 - 330 x + 315 x^2 - 160 x^3 + 45 x^4 - 6 x^5
= (3 - 2 x) (6 - 3 x + x^2) (8 - 9 x + 3 x^2)
377 - 1008 x + 1155 x^2 - 735 x^3 + 280 x^4 - 63 x^5 + 7 x^6
987 - 3016 x + 4032 x^2 - 3080 x^3 + 1470 x^4 - 448 x^5 + 84 x^6 - 8 x^7
= (3 - 2 x) (7 - 6 x + 2 x^2) (47 - 72 x + 42 x^2 - 12 x^3 + 2 x^4)
-
g[x_, n_] := Numerator[ Factor[D[1/(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}] (* A328645 array *)
Showing 1-2 of 2 results.
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