A303941 -id:A303941 - 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

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

A318184 a(n) = 2^(n * (n - 1)/2) * 3^((n - 1) * (n - 2)) * n^(n - 3).

Original entry on oeis.org

1, 1, 72, 186624, 13604889600, 24679069470425088, 1036715783690392172494848, 962459606796748852884396910313472, 19112837387997044228759204010262201783812096, 7926475921550134182551017087135940323782552453120000000, 67406870957147550175650545441605700298239194363455522532832462241792

Offset: 1

Rigoberto Florez, Aug 20 2018

nonn

Discriminant of Fermat polynomials.

F(0)=0, F(1)=1 and F(n) = 3x F(n - 1) -2 F(n - 2) if n>1.

Muniru A Asiru, Table of n, a(n) for n = 1..39
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Fermat Polynomial

Cf. A193678, A007701, A007701, A193678, A303941.

seq(2^(n*(n-1)/2)*3^((n-1)*(n-2))*n^(n-3),n=1..12); # Muniru A Asiru, Dec 07 2018

F[0] = 0; F[1] = 1; F[n_] := F[n] = 3 x F[n - 1] - 2 F[n - 2];
a[n_] := Discriminant[F[n], x];
Array[a, 11] (* Jean-François Alcover, Dec 07 2018 *)

a(n) = 2^(n*(n-1)/2) * 3^((n-1)*(n-2)) * n^(n-3); \\ Michel Marcus, Dec 07 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).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A318184 a(n) = 2^(n * (n - 1)/2) * 3^((n - 1) * (n - 2)) * n^(n - 3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

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.