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%I #18 Jun 28 2022 03:59:49
%S 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,
%T 2695,7084,0,1,19,184,1196,5750,20930,53820,0,1,22,247,1872,10647,
%U 47502,166257,420732,0,1,25,319,2755,17980,93496,395560,1344904,3362260
%N The Fuss-Catalan triangle of order 3, read by rows. Related to quartic trees.
%C 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.)
%C 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.
%F The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
%F 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).
%F T(n, k) = (3*(n - k) + 4)*(3*n + k - 1)!/((3*n + 1)!*(k - 1)!) for k > 0; T(n, 0) = n^0.
%F 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:
%F T(n, k) = [x^k] (0^n + (x - 4*x^2)/(1 - x)^(3*n + 2)).
%e Table T(n, k) begins:
%e [0] [1]
%e [1] [0, 1]
%e [2] [0, 1, 4]
%e [3] [0, 1, 7, 22]
%e [4] [0, 1, 10, 49, 140]
%e [5] [0, 1, 13, 85, 357, 969]
%e [6] [0, 1, 16, 130, 700, 2695, 7084]
%e [7] [0, 1, 19, 184, 1196, 5750, 20930, 53820]
%e [8] [0, 1, 22, 247, 1872, 10647, 47502, 166257, 420732]
%e [9] [0, 1, 25, 319, 2755, 17980, 93496, 395560, 1344904, 3362260]
%e .
%e Seen as an array reading the diagonals starting from the main diagonal:
%e [0] 1, 1, 4, 22, 140, 969, 7084, 53820, 420732, ... A002293
%e [1] 0, 1, 7, 49, 357, 2695, 20930, 166257, 1344904, ... A233658
%e [2] 0, 1, 10, 85, 700, 5750, 47502, 395560, 3321120, ... A233667
%e [3] 0, 1, 13, 130, 1196, 10647, 93496, 816816, 7128420, ...
%e [4] 0, 1, 16, 184, 1872, 17980, 167552, 1535352, 13934752, ...
%o (Python)
%o from functools import cache
%o from itertools import accumulate
%o @cache
%o def Trow(n: int) -> list[int]:
%o if n == 0: return [1]
%o if n == 1: return [0, 1]
%o row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
%o return list(accumulate(accumulate(accumulate(row))))
%o for n in range(11): print(Trow(n))
%Y A002293 (main diagonal), A233658 (subdiagonal), A233667 (diagonal 2), A016777 (column 2), A196678 (row sums).
%Y Cf. A123110 (triangle of order 0), A355173 (triangle of order 1), A355172 (triangle of order 2), A355262 (Fuss-Catalan array).
%K nonn,tabl
%O 0,6
%A _Peter Luschny_, Jun 25 2022