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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.
5
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
0,2
COMMENTS
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.
REFERENCES
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.
EXAMPLE
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
MATHEMATICA
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
PROG
(PARI) 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
CROSSREFS
Row sums give A014983.
Sequence in context: A361640 A323506 A357988 * A193949 A246679 A244153
KEYWORD
tabf,easy,sign
AUTHOR
Zagros Lalo, May 04 2018
STATUS
approved