A302747 -id:A302747 - cp's OEIS frontend

A303901 Triangle read by rows of coefficients in expansion of (3-2x)^n, where n is a nonnegative integer.

1, 3, -2, 9, -12, 4, 27, -54, 36, -8, 81, -216, 216, -96, 16, 243, -810, 1080, -720, 240, -32, 729, -2916, 4860, -4320, 2160, -576, 64, 2187, -10206, 20412, -22680, 15120, -6048, 1344, -128, 6561, -34992, 81648, -108864, 90720, -48384, 16128, -3072, 256, 19683, -118098, 314928, -489888, 489888, -326592, 145152, -41472, 6912, -512

Offset: 0

Zagros Lalo, May 02 2018

This is a signed version of A038220.
Row n gives coefficients in expansion of (3-2x)^n.
The numbers in rows of triangles in A302747 and A303941 are along skew diagonals pointing top-left and top-right in center-justified triangle of coefficients in expansions of (3-2x)^n (A303901).
This is the lower triangular Riordan matrix (1/(1-3*t), -2*t/(1-3*t)), hence a convolution matrix. See the g.f.s. - Wolfdieter Lang, Jun 28 2018

			Triangle begins:
  n \k 0     1       2       3      4      5       6      7      8    9  ...
  --------------------------------------------------------------------------
  0 |  1
  1 |  3     -2
  2 |  9     -12     4
  3 |  27    -54     36     -8
  4 |  81    -216    216    -96     16
  5 |  243   -810    1080   -720    240    -32
  6 |  729   -2916   4860   -4320   2160   -576    64
  7 |  2187  -10206  20412  -22680  15120  -6048   1344   -128
  8 |  6561  -34992  81648  -108864 90720  -48384  16128  -3072  256
  9 |  19683 -118098 314928 -489888 489888 -326592 145152 -41472 6912 -512

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 394, 396, 398.

Zagros Lalo, Center-justified Triangle
Zagros Lalo, Skew Diagonals in center-justified Triangle

Cf. A013620 (unsigned), A000012 (row sums), A000351 (alternating row sums).

For[i = 0, i < 4, i++, Print[CoefficientList[Expand[(3 - 2 x)^i],x]]]

T(0,0) = 1; T(n,k) = 3*T(n-1,k) - 2*T(n-1,k-1) for k = 0,1,...,n; T(n,k)=0 for n or k < 0.
G.f. of row polynomials: 1 / (1 - 3*t + 2*t*x).
G.f. of column k: (-2*x)^k/(1-3*x)^(k+1), for k >= 0.

Edited - Wolfdieter Lang, Jun 28 2018

A317498 Triangle read by rows of coefficients in expansions of (-2 + 3*x)^n, where n is nonnegative integer.

Original entry on oeis.org

1, -2, 3, 4, -12, 9, -8, 36, -54, 27, 16, -96, 216, -216, 81, -32, 240, -720, 1080, -810, 243, 64, -576, 2160, -4320, 4860, -2916, 729, -128, 1344, -6048, 15120, -22680, 20412, -10206, 2187, 256, -3072, 16128, -48384, 90720, -108864, 81648, -34992, 6561, -512, 6912, -41472, 145152, -326592, 489888, -489888, 314928, -118098, 19683

Offset: 0

Zagros Lalo, Jul 31 2018

Row n gives coefficients in expansion of (-2 + 3*x)^n.
This is a signed version of A013620.
The coefficients in the expansion of 1/(1-x) are given by the sequence generated by the row sums.
The row sums give A000012 (The simplest sequence of positive numbers: the all 1's sequence).
The numbers in rows of triangles in A302747 and A303941 (Triangle of coefficients of Fermat polynomials) are along first layer skew diagonals pointing top-right and top-left in center-justified triangle of coefficients in expansions of (-2 + 3*x)^n, see links.

			Triangle begins:
     1;
    -2,     3;
     4,   -12,      9;
    -8,    36,    -54,     27;
    16,   -96,    216,   -216,      81;
   -32,   240,   -720,   1080,    -810,     243;
    64,  -576,   2160,  -4320,    4860,   -2916,     729;
  -128,  1344,  -6048,  15120,  -22680,   20412,  -10206,   2187;
   256, -3072,  16128, -48384,   90720, -108864,   81648, -34992,    6561;
  -512,  6912, -41472, 145152, -326592,  489888, -489888, 314928, -118098, 19683;
  ...

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

Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (-2 + 3 x)^n.
Eric Weisstein's World of Mathematics, Fermat Polynomial.

Row sums give A000012.
Cf. A013620 ((2+3*x)^n).
Cf. A302747, A303941.

Flat(List([0..8],n->List([0..n],k->(-2)^(n-k)*3^k/(Factorial(n-k)*Factorial(k))*Factorial(n)))); # Muniru A Asiru, Aug 01 2018

t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, -2 t[n - 1, k] + 3 t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
t[n_, k_] := t[n, k] = ((-2)^(n - k) 3^k)/((n - k)! k!) n!;Table[t[n, k], {n, 0, 9}, {k, 0, n} ] // Flatten
Table[CoefficientList[(-2 + 3 x)^n, x], {n, 0, 9}] // Flatten

trianglerows(n) = my(v=[]); for(k=0, n-1, v=Vec((-2+3*x)^k + O(x^(k+1))); print(v))
/* Print initial 10 rows of triangle as follows */
trianglerows(10) \\ Felix Fröhlich, Jul 31 2018

T(0,0) = 1; T(n,k) = -2 * T(n-1,k) + 3 * T(n-1,k-1) for k = 0,1,...,n and T(n,k)=0 for n or k < 0.
T(n, k) = ((-2)^(n - k) 3^k)/((n - k)! k!) n! for k = 0,1..n.
G.f.: 1 / (1 + 2*x - 3*x*t).

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).

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A303901 Triangle read by rows of coefficients in expansion of (3-2x)^n, where n is a nonnegative integer.

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A317498 Triangle read by rows of coefficients in expansions of (-2 + 3*x)^n, where n is nonnegative integer.

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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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