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A303901
Triangle read by rows of coefficients in expansion of (3-2x)^n, where n is a nonnegative integer.
6
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
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.
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
Cf. A013620 (unsigned), A000012 (row sums), A000351 (alternating row sums).
Sequence in context: A366348 A274827 A038220 * A053151 A053088 A077898
KEYWORD
tabl,easy,sign
AUTHOR
Zagros Lalo, May 02 2018
EXTENSIONS
Edited - Wolfdieter Lang, Jun 28 2018
STATUS
approved