A355174 The Fuss-Catalan triangle of order 3, read by rows. Related to quartic trees.

1, 0, 1, 0, 1, 4, 0, 1, 7, 22, 0, 1, 10, 49, 140, 0, 1, 13, 85, 357, 969, 0, 1, 16, 130, 700, 2695, 7084, 0, 1, 19, 184, 1196, 5750, 20930, 53820, 0, 1, 22, 247, 1872, 10647, 47502, 166257, 420732, 0, 1, 25, 319, 2755, 17980, 93496, 395560, 1344904, 3362260

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.)

This definition also includes the Fuss-Catalan numbers A002293(n) = T(n, n), row 4 in A355262. For m = 1 see A355173 and for m = 2 A355172. More generally, for n >= 1 all Fuss-Catalan sequences (A355262(n, k), k >= 0) are the main diagonals of the Fuss-Catalan triangles of order n - 1.

Links

Table of n, a(n) for n=0..54.

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

A002293 (main diagonal), A233658 (subdiagonal), A233667 (diagonal 2), A016777 (column 2), A196678 (row sums).

Cf. A123110 (triangle of order 0), A355173 (triangle of order 1), A355172 (triangle of order 2), A355262 (Fuss-Catalan array).

Sequence in context: A334385 A201560 A255644 * A059678 A079642 A342911

Adjacent sequences: A355171 A355172 A355173 * A355175 A355176 A355177

Keyword

nonn,tabl

Author

Peter Luschny, Jun 25 2022

Status

approved