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A355172
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The Fuss-Catalan triangle of order 2, read by rows. Related to ternary trees.
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3
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1, 0, 1, 0, 1, 3, 0, 1, 5, 12, 0, 1, 7, 25, 55, 0, 1, 9, 42, 130, 273, 0, 1, 11, 63, 245, 700, 1428, 0, 1, 13, 88, 408, 1428, 3876, 7752, 0, 1, 15, 117, 627, 2565, 8379, 21945, 43263, 0, 1, 17, 150, 910, 4235, 15939, 49588, 126500, 246675
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OFFSET
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0,6
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COMMENTS
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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 = 2. (See the Python program for a reference implementation.)
This definition also includes the Fuss-Catalan numbers A001764(n) = T(n, n), or row 3 in A355262. For m = 1 see A355173 and for m = 3 A355174. 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.
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LINKS
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FORMULA
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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, 2).
T(n, k) = (2*n - 2*k + 3)*(2*n + k - 1)!/((2*n + 1)!*(k - 1)!) for k > 0; T(n, 0) = 0^n.
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 - 3*x^2)/(1 - x)^(2*n + 2)).
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EXAMPLE
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Table T(n, k) begins:
[0] [1]
[1] [0, 1]
[2] [0, 1, 3]
[3] [0, 1, 5, 12]
[4] [0, 1, 7, 25, 55]
[5] [0, 1, 9, 42, 130, 273]
[6] [0, 1, 11, 63, 245, 700, 1428]
[7] [0, 1, 13, 88, 408, 1428, 3876, 7752]
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Seen as an array reading the diagonals starting from the main diagonal:
[0] 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, ... A001764
[1] 0, 1, 5, 25, 130, 700, 3876, 21945, 126500, 740025, ... A102893
[2] 0, 1, 7, 42, 245, 1428, 8379, 49588, 296010, 1781325, ... A102594
[3] 0, 1, 9, 63, 408, 2565, 15939, 98670, 610740, 3786588, ... A230547
[4] 0, 1, 11, 88, 627, 4235, 27830, 180180, 1157013, 7396972, ...
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PROG
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(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(row)))
for n in range(9): print(Trow(n))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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