A305834 Triangle read by rows: T(0,0)= 1; T(n,k)= T(n-1,k) + 4*T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.
1, 1, 1, 4, 1, 8, 1, 12, 16, 1, 16, 48, 1, 20, 96, 64, 1, 24, 160, 256, 1, 28, 240, 640, 256, 1, 32, 336, 1280, 1280, 1, 36, 448, 2240, 3840, 1024, 1, 40, 576, 3584, 8960, 6144, 1, 44, 720, 5376, 17920, 21504, 4096
Offset: 0
Examples
Triangle begins: 1; 1; 1, 4; 1, 8; 1, 12, 16; 1, 16, 48; 1, 20, 96, 64; 1, 24, 160, 256; 1, 28, 240, 640, 256; 1, 32, 336, 1280, 1280; 1, 36, 448, 2240, 3840, 1024; 1, 40, 576, 3584, 8960, 6144; 1, 44, 720, 5376, 17920, 21504, 4096; 1, 48, 880, 7680, 32256, 57344, 28672; 1, 52, 1056, 10560, 53760, 129024, 114688, 16384; 1, 56, 1248, 14080, 84480, 258048, 344064, 131072; 1, 60, 1456, 18304, 126720, 473088, 860160, 589824, 65536; 1, 64, 1680, 23296, 183040, 811008, 1892352, 1966080, 589824;
References
- Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 371, 372.
Links
- Shara Lalo, Right justified triangle
- Shara Lalo, Skew diagonals in triangle A013611
Crossrefs
Programs
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Mathematica
t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 4 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten
Formula
G.f.: 1/(1 - t*x - 4*t^2).
Column k is binomial (n + k - 1, k) * 4^k.
Comments