A327315
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 (x-2)/(x^2-x+1)).
Original entry on oeis.org
-2, 1, -1, 4, -1, 1, 3, -6, 1, 2, -4, -6, 8, -1, 1, -10, 10, 10, -10, 1, -1, -6, 30, -20, -15, 12, -1, -2, 7, 21, -70, 35, 21, -14, 1, -1, 16, -28, -56, 140, -56, -28, 16, -1, 1, 9, -72, 84, 126, -252, 84, 36, -18, 1, 2, -10, -45, 240, -210, -252, 420, -120
Offset: 0
First eight rows:
-2, 1;
-1, 4, -1;
1, 3, -6, 1;
2, -4, -6, 8, -1;
1, -10, 10, 10, -10, 1;
-1, -6, 30, -20, -15, 12, -1;
-2, 7, 21, -70, 35, 21, -14, 1;
-1, 16, -28, -56, 140, -56, -28, 16, -1;
First eight polynomials:
-2 + x
-1 + 4 x - x^2
1 + 3 x - 6 x^2 + x^3
2 - 4 x - 6 x^2 + 8 x^3 - x^4
(1 + x) (1 - 11 x + 21 x^2 - 11 x^3 + x^4)
-1 - 6 x + 30 x^2 - 20 x^3 - 15 x^4 + 12 x^5 - x^6
(-2 + x) (1 - 3 x - 12 x^2 + 29 x^3 - 3 x^4 - 12 x^5 + x^6)
-1 + 16 x - 28 x^2 - 56 x^3 + 140 x^4 - 56 x^5 - 28 x^6 + 16 x^7 - x^8
-
g[x_, n_] := Numerator[ Factor[D[(x - 2)/(x^2 - x + 1), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* A327315 polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x];
Table[h[n], {n, 0, 10}] (* A327315 sequence *)
Column[%] (* A327315 array *)
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)).
Original entry on oeis.org
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
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
-
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 *)
A328649
Irregular triangular array read by rows: row n shows the coefficients of the following polynomial of degree n: (1/n!)*(numerator of n-th derivative of (x-2)/(1-x-x^2)).
Original entry on oeis.org
2, -1, -1, -4, 1, 3, 3, 6, -1, -4, -12, -6, -8, 1, 7, 20, 30, 10, 10, -1, -11, -42, -60, -60, -15, -12, 1, 18, 77, 147, 140, 105, 21, 14, -1, -29, -144, -308, -392, -280, -168, -28, -16, 1, 47, 261, 648, 924, 882, 504, 252, 36, 18, -1, -76, -470, -1305
Offset: 0
First eight rows:
2, -1;
-1, -4, 1;
3, 3, 6, -1;
-4, -12, -6, -8, 1;
7, 20, 30, 10, 10, -1;
-11, -42, -60, -60, -15, -12, 1;
18, 77, 147, 140, 105, 21, 14, -1;
-29, -144, -308, -392, -280, -168, -28, -16, 1;
First eight polynomials:
2 - x
-1 - 4 x + x^2
3 + 3 x + 6 x^2 - x^3
-4 - 12 x - 6 x^2 - 8 x^3 + x^4
7 + 20 x + 30 x^2 + 10 x^3 + 10 x^4 - x^5
-11 - 42 x - 60 x^2 - 60 x^3 - 15 x^4 - 12 x^5 + x^6
18 + 77 x + 147 x^2 + 140 x^3 + 105 x^4 + 21 x^5 + 14 x^6 - x^7
-29 - 144 x - 308 x^2 - 392 x^3 - 280 x^4 - 168 x^5 - 28 x^6 - 16 x^7 + x^8
-
g[x_, n_] := Numerator[ Factor[D[(x - 2)/(1 - x - x^2), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* A328649 polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x];
Table[h[n], {n, 0, 10}] (* A328649 sequence *)
Column[%] (* A328649 array *)
A328650
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-2x^2).
Original entry on oeis.org
-1, 1, 4, -3, -6, -12, 5, 24, 24, 32, -11, -50, -120, -80, -80, 21, 132, 300, 480, 240, 192, -43, -294, -924, -1400, -1680, -672, -448, 85, 688, 2352, 4928, 5600, 5376, 1792, 1024, -171, -1530, -6192, -14112, -22176, -20160, -16128, -4608, -2304, 341, 3420
Offset: 0
First eight rows:
-1;
1, 4;
3, -6, -12;
5, 24, 24, 32;
-11, -50, -120, -80, -80;
21, 132, 300, 480, 240, 192;
-43, -294, -924, -1400, -1680, -672, -448;
85, 688, 2352, 4928, 5600, 5376, 1792, 1024;
First eight polynomials:
-1
1 + 4 x
-3 (1 + 2 x + 4 x^2)
(1 + 4 x) (5 + 4 x + 8 x^2)
-11 - 50 x - 120 x^2 - 80 x^3 - 80 x^4
3 (1 + 4 x) (1 + 2 x + 4 x^2) (7 + 2 x + 4 x^2)
-43 - 294 x - 924 x^2 - 1400 x^3 - 1680 x^4 - 672 x^5 - 448 x^6
(1 + 4 x) (5 + 4 x + 8 x^2) (17 + 56 x + 120 x^2 + 32 x^3 + 32 x^4)
-
g[x_, n_] := Numerator[ Factor[D[1/(1 - x - 2 x^2), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* A328650 polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x];
Table[h[n], {n, 0, 10}] (* A328650 sequence *)
Column[%] (* A328650 array *)
Showing 1-4 of 4 results.
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