A303901 -id:A303901 - cp's OEIS frontend

A303941 Triangle read by rows: T(0,0) = 1; T(n,k) = 3T(n-1,k) - 2T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. Triangle of coefficients of Fermat polynomials.

1, 3, 9, -2, 27, -12, 81, -54, 4, 243, -216, 36, 729, -810, 216, -8, 2187, -2916, 1080, -96, 6561, -10206, 4860, -720, 16, 19683, -34992, 20412, -4320, 240, 59049, -118098, 81648, -22680, 2160, -32, 177147, -393660, 314928, -108864, 15120, -576, 531441, -1299078, 1180980, -489888, 90720, -6048, 64

Offset: 0

Zagros Lalo, May 03 2018

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A303901 ((3-2x)^n).
Row n gives coefficients of Fermat polynomial.
The coefficients in the expansion of 1/(1-3x+2x^2) are given by the sequence generated by the row sums.

			Triangle begins:
n\k |       0         1        2         3        4        5      6     7
----+--------------------------------------------------------------------
   0|       1
   1|       3
   2|       9        -2
   3|      27       -12
   4|      81       -54        4
   5|     243      -216       36
   6|     729      -810      216        -8
   7|    2187     -2916     1080       -96
   8|    6561    -10206     4860      -720       16
   9|   19683    -34992    20412     -4320      240
  10|   59049   -118098    81648    -22680     2160      -32
  11|  177147   -393660   314928   -108864    15120     -576
  12|  531441  -1299078  1180980   -489888    90720    -6048     64
  13| 1594323  -4251528  4330260  -2099520   489888   -48384   1344
  14| 4782969 -13817466 15588936  -8660520  2449440  -326592  16128  -128
  15|14348907 -44641044 55269864 -34642080 11547360 -1959552 145152 -3072

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 104, 394, 395.

Zagros Lalo, Left-justified triangle
Eric Weisstein's World of Mathematics, Fermat Polynomial.

Row sums give A000225.
Some row sums give A001348.
Cf. A303901.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 3 t[n - 1, k] - 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 14}, {k, 0, Floor[n/2]}] // Flatten

T(n,k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 3*T(n-1,k) - 2*T(n-2,k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 10 2018

A302747 Triangle read by rows: T(0,0) = 1; T(n,k) = -2T(n-1,k) + 3T(n-2,k-1) for 0 <= k <= floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, -2, 4, 3, -8, -12, 16, 36, 9, -32, -96, -54, 64, 240, 216, 27, -128, -576, -720, -216, 256, 1344, 2160, 1080, 81, -512, -3072, -6048, -4320, -810, 1024, 6912, 16128, 15120, 4860, 243, -2048, -15360, -41472, -48384, -22680, -2916, 4096, 33792, 103680, 145152, 90720, 20412, 729, -8192, -73728, -253440

Offset: 0

Zagros Lalo, May 04 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle in A303901 ((3-2x)^n).

The coefficients in the expansion of 1/(1-3x+2x^2) are given by the sequence generated by the row sums.

			Triangle begins:
.
   n | k = 0      1       2       3       4       5       6
  ---+-----------------------------------------------------
   0 |     1
   1 |    -2
   2 |     4      3
   3 |    -8    -12
   4 |    16     36       9
   5 |   -32    -96     -54
   6 |    64    240     216      27
   7 |  -128   -576    -720    -216
   8 |   256   1344    2160    1080      81
   9 |  -512  -3072   -6048   -4320    -810
  10 |  1024   6912   16128   15120    4860     243
  11 | -2048 -15360  -41472  -48384  -22680   -2916
  12 |  4096  33792  103680  145152   90720   20412     729
  13 | -8192 -73728 -253440 -414720 -326592 -108864  -10206

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 394-396.

Zagros Lalo, Left-justified triangle

Row sums give A014983.

Cf. A303901, A303941.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, -2 t[n - 1, k] + 3 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

T(n,k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, -2*T(n-1,k) + 3*T(n-2,k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 10 2018

A317499 Coefficients in expansion of 1/(1 + 2x - 3x^3).

Original entry on oeis.org

1, -2, 4, -5, 4, 4, -23, 58, -104, 139, -104, -104, 625, -1562, 2812, -3749, 2812, 2812, -16871, 42178, -75920, 101227, -75920, -75920, 455521, -1138802, 2049844, -2733125, 2049844, 2049844, -12299063, 30747658, -55345784, 73794379, -55345784, -55345784

Offset: 0

Zagros Lalo, Jul 31 2018

The coefficients in the expansion of 1/(1 + 2*x - 3*x^3) are given by the sequence generated by the row sums in triangle A317503.

Coefficients in expansion of 1/(1 + 2*x - 3*x^3) are given by the sum of numbers along second Layer skew diagonals pointing top-left in triangle A303901 ((3-2*x)^n) and by the sum of numbers along second Layer skew diagonals pointing top-right in triangle A317498 ((-2+3*x)^n), see links.

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 396, 397.

Colin Barker, Table of n, a(n) for n = 0..1000
Zagros Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (3 - 2x)^n
Zagros Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (-2 + 3x)^n
Index entries for linear recurrences with constant coefficients, signature (-2,0,3).

Cf. A317502, A317503.

Cf. A303901, A317498.

a:=[1,-2,4];; for n in [4..40] do a[n]:=-2*a[n-1]+3*a[n-3]; od; a; # Muniru A Asiru, Aug 01 2018

seq(coeff(series(1/(1+2*x-3*x^3), x,n+1),x,n),n=0..40); # Muniru A Asiru, Aug 01 2018

CoefficientList[Series[1/(1 + 2 x - 3 x^3), {x, 0, 40}], x]
a[0] = 1; a[n_] := a[n] = If[n < 0, 0, -2 * a[n - 1] + 3 * a[n - 3]]; Table[a[n], {n, 0, 40}] // Flatten
LinearRecurrence[{-2, 0, 3}, {1, -2, 4}, 41]

Vec(1 / ((1 - x)*(1 + 3*x + 3*x^2)) + O(x^40)) \\ Colin Barker, Aug 02 2018

a(0)=1, a(n) = -2*a(n-1) + 3*a(n-3) for n = 0,1...; a(n)=0 for n < 0.

a(n) = (2^(-n)*(2^n + (-3-i*sqrt(3))^n*(3-2*i*sqrt(3)) + (-3+i*sqrt(3))^n*(3+2*i*sqrt(3)))) / 7 where i=sqrt(-1). - Colin Barker, Aug 02 2018

A317502 Triangle read by rows: T(0,0) = 1; T(n,k) = 3 T(n-1,k) - 2 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 3, 9, 27, -2, 81, -12, 243, -54, 729, -216, 4, 2187, -810, 36, 6561, -2916, 216, 19683, -10206, 1080, -8, 59049, -34992, 4860, -96, 177147, -118098, 20412, -720, 531441, -393660, 81648, -4320, 16, 1594323, -1299078, 314928, -22680, 240, 4782969, -4251528, 1180980, -108864, 2160

Offset: 0

Shara Lalo, Aug 02 2018

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-right in center-justified triangle given in A303901 ((3-2*x)^n) and along "second layer" skew diagonals pointing top-left in center-justified triangle given in A317498 ((-2+3x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (3-2*x)^n and (-2+3x)^n are given in A303941 and A302747 respectively.) The coefficients in the expansion of 1/(1-3x+2x^3) are given by the sequence generated by the row sums. The row sums give A077846. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.7320508075688772... (A090388: 1+sqrt(3)), when n approaches infinity.

			Triangle begins:
        1;
        3;
        9;
        27,        -2;
        81,       -12;
       243,       -54;
       729,      -216,        4;
      2187,      -810,       36;
      6561,     -2916,      216;
     19683,    -10206,     1080,       -8;
     59049,    -34992,     4860,      -96;
    177147,   -118098,    20412,     -720;
    531441,   -393660,    81648,    -4320,    16;
   1594323,  -1299078,   314928,   -22680,   240;
   4782969,  -4251528,  1180980,  -108864,  2160;
  14348907, -13817466,  4330260,  -489888, 15120,  -32;
  43046721, -44641044, 15588936, -2099520, 90720, -576;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 136, 396, 397.

Row sums give A077846.
Cf. A303901, A317498.
Cf. A090388.
Cf. A303941, A302747.

t[n_, k_] := t[n, k] = 3^(n - 3k) * (-2)^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 3 * t[n - 1, k] - 2 * t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = 3^(n - 3k) * (-2)^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

A317503 Triangle read by rows: T(0,0) = 1; T(n,k) = -2 T(n-1,k) + 3 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, -2, 4, -8, 3, 16, -12, -32, 36, 64, -96, 9, -128, 240, -54, 256, -576, 216, -512, 1344, -720, 27, 1024, -3072, 2160, -216, -2048, 6912, -6048, 1080, 4096, -15360, 16128, -4320, 81, -8192, 33792, -41472, 15120, -810, 16384, -73728, 103680, -48384, 4860, -32768, 159744, -253440, 145152, -22680, 243

Offset: 0

Shara Lalo, Aug 02 2018

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-left in center-justified triangle given in A303901 ((3-2*x)^n) and along "second layer" skew diagonals pointing top-right in center-justified triangle given in A317498 ((-2+3x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (3-2*x)^n and (-2+3x)^n are given in A303941 and A302747 respectively.) The coefficients in the expansion of 1/(1 + 2x - 3x^3) are given by the sequence generated by the row sums. The row sums give A317499.

			Triangle begins:
       1;
      -2;
       4;
      -8,       3;
      16,     -12;
     -32,      36;
      64,     -96,       9;
    -128,     240,     -54;
     256,    -576,     216;
    -512,    1344,    -720,      27;
    1024,   -3072,    2160,    -216;
   -2048,    6912,   -6048,    1080;
    4096,  -15360,   16128,   -4320,     81;
   -8192,   33792,  -41472,   15120,   -810;
   16384,  -73728,  103680,  -48384,   4860;
  -32768,  159744, -253440,  145152, -22680,   243;
   65536, -344064,  608256, -414720,  90720, -2916;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 136, 396, 397.

Row sums give A317499.
Cf. A303901, A317498.
Cf. A090388.
Cf. A303941, A302747.

t[n_, k_] := t[n, k] = (-2)^(n - 3k) * 3^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, -2 * t[n - 1, k] + 3 * t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = (-2)^(n - 3k) * 3^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

cp's OEIS Frontend

A303941 Triangle read by rows: T(0,0) = 1; T(n,k) = 3*T(n-1,k) - 2*T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. Triangle of coefficients of Fermat polynomials.

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A302747 Triangle read by rows: T(0,0) = 1; T(n,k) = -2*T(n-1,k) + 3*T(n-2,k-1) for 0 <= k <= floor(n/2); T(n,k)=0 for n or k < 0.

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A317499 Coefficients in expansion of 1/(1 + 2*x - 3*x^3).

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A317502 Triangle read by rows: T(0,0) = 1; T(n,k) = 3 T(n-1,k) - 2 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

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A317503 Triangle read by rows: T(0,0) = 1; T(n,k) = -2 T(n-1,k) + 3 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

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A303941 Triangle read by rows: T(0,0) = 1; T(n,k) = 3T(n-1,k) - 2T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. Triangle of coefficients of Fermat polynomials.

A302747 Triangle read by rows: T(0,0) = 1; T(n,k) = -2T(n-1,k) + 3T(n-2,k-1) for 0 <= k <= floor(n/2); T(n,k)=0 for n or k < 0.

A317499 Coefficients in expansion of 1/(1 + 2x - 3x^3).