OFFSET
0,2
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
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
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 - 3t + 2t x).G.f. of column k: (-2*x)^k/(1-3*x)^(k+1), for k >= 0.
EXAMPLE
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
…
MATHEMATICA
For[i = 0, i < 4, i++, Print[CoefficientList[Expand[(3 - 2 x)^i], x]]]
CROSSREFS
KEYWORD
AUTHOR
Zagros Lalo, May 02 2018
EXTENSIONS
Edited - Wolfdieter Lang, Jun 28 2018
STATUS
approved