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

Author

Zagros Lalo, May 02 2018

Keywords

• tabl
• easy
• sign

Comments

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

Examples

			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
		

References

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

Links

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

Crossrefs

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

Programs

Mathematica

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

Formula

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.

Extensions

Edited - Wolfdieter Lang, Jun 28 2018