OFFSET
0,6
COMMENTS
The Fuss-Catalan triangle of order m is a regular, (0, 0)-based table recursively defined as follows: Set row(0) = [1] and row(1) = [0, 1]. Now assume row(n-1) already constructed and duplicate the last element of row(n-1). Next apply the cumulative sum m times to this list to get row(n). Here m = 3. (See the Python program for a reference implementation.)
FORMULA
The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
FCT(n, k, m) = (m*(n - k) + m + 1)*(m*n + k - 1)!/((m*n + 1)!*(k - 1)!)) for k > 0 and FCT(n, 0, m) = 0^n. The case considered here is T(n, k) = FCT(n, k, 3).
T(n, k) = (3*(n - k) + 4)*(3*n + k - 1)!/((3*n + 1)!*(k - 1)!) for k > 0; T(n, 0) = n^0.
The g.f. of row n of the FC-triangle of order m is 0^n + (x - (m + 1)*x^2) / (1 - x)^(m*n + 2), thus:
T(n, k) = [x^k] (0^n + (x - 4*x^2)/(1 - x)^(3*n + 2)).
EXAMPLE
Table T(n, k) begins:
[0] [1]
[1] [0, 1]
[2] [0, 1, 4]
[3] [0, 1, 7, 22]
[4] [0, 1, 10, 49, 140]
[5] [0, 1, 13, 85, 357, 969]
[6] [0, 1, 16, 130, 700, 2695, 7084]
[7] [0, 1, 19, 184, 1196, 5750, 20930, 53820]
[8] [0, 1, 22, 247, 1872, 10647, 47502, 166257, 420732]
[9] [0, 1, 25, 319, 2755, 17980, 93496, 395560, 1344904, 3362260]
.
Seen as an array reading the diagonals starting from the main diagonal:
[0] 1, 1, 4, 22, 140, 969, 7084, 53820, 420732, ... A002293
[1] 0, 1, 7, 49, 357, 2695, 20930, 166257, 1344904, ... A233658
[2] 0, 1, 10, 85, 700, 5750, 47502, 395560, 3321120, ... A233667
[3] 0, 1, 13, 130, 1196, 10647, 93496, 816816, 7128420, ...
[4] 0, 1, 16, 184, 1872, 17980, 167552, 1535352, 13934752, ...
PROG
(Python)
from functools import cache
from itertools import accumulate
@cache
def Trow(n: int) -> list[int]:
if n == 0: return [1]
if n == 1: return [0, 1]
row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
return list(accumulate(accumulate(accumulate(row))))
for n in range(11): print(Trow(n))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jun 25 2022
STATUS
approved