A317503 Triangle read by rows: T(0,0) = 1; T(n,k) = -2 T(n-1,k) + 3 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.
1, -2, 4, -8, 3, 16, -12, -32, 36, 64, -96, 9, -128, 240, -54, 256, -576, 216, -512, 1344, -720, 27, 1024, -3072, 2160, -216, -2048, 6912, -6048, 1080, 4096, -15360, 16128, -4320, 81, -8192, 33792, -41472, 15120, -810, 16384, -73728, 103680, -48384, 4860, -32768, 159744, -253440, 145152, -22680, 243
Offset: 0
Examples
Triangle begins:
1;
-2;
4;
-8, 3;
16, -12;
-32, 36;
64, -96, 9;
-128, 240, -54;
256, -576, 216;
-512, 1344, -720, 27;
1024, -3072, 2160, -216;
-2048, 6912, -6048, 1080;
4096, -15360, 16128, -4320, 81;
-8192, 33792, -41472, 15120, -810;
16384, -73728, 103680, -48384, 4860;
-32768, 159744, -253440, 145152, -22680, 243;
65536, -344064, 608256, -414720, 90720, -2916;
References
- Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 136, 396, 397.
Links
Programs
-
Mathematica
t[n_, k_] := t[n, k] = (-2)^(n - 3k) * 3^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]} ] // Flatten t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, -2 * t[n - 1, k] + 3 * t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]}] // Flatten
Formula
T(n,k) = (-2)^(n - 3k) * 3^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).
Comments