OFFSET
0,2
COMMENTS
This is Boethius's triangle, with rows read as columns. See the link and reference.
REFERENCES
Thomas Sonar, 3000 Jahre Analysis, 2. Auflage, Springer Spektrum, 2016, p.94, Abb. 3.1.2 und Abb. 3.1.3.
LINKS
Anicius Manlius Severinus Boethius, De Institutione Arithmetica, 1488, p. 55 of 104, top of left column.
FORMULA
T(n, k) = 3^(n-k)*4^k, for n >= 0, and k = 1, 2, ..., n.
G.f. of row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k: G(x, y) = 1/((1 - 3*x)*(1 - 4*x*y)).
EXAMPLE
The triangle T begins:
n\k 0 1 2 3 4 5 6 7 8 9 ...
0: 1
1: 3 4
2: 9 12 16
3: 27 36 48 64
4: 81 108 144 192 256
5: 243 324 432 576 768 1024
6: 729 972 1296 1728 2304 3072 4096
7: 2187 2916 3888 5184 6912 9216 12288 16384
8: 6561 8748 11664 15552 20736 27648 36864 49152 65536
9: 19683 26244 34992 46656 62208 82944 110592 147456 196608 262144
...
MATHEMATICA
T[n_, k_] := 3^(n - k) * 4^k; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 05 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Aug 01 2022
STATUS
approved