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.

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

Author

Zagros Lalo, May 04 2018

Keywords

• tabf
• easy
• sign

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.

Examples

			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
		

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.

Links

• Zagros Lalo, Left-justified triangle

Crossrefs

Row sums give A014983.

Cf. A303901, A303941.

Programs

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

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