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

%I A355174 #21 Mar 22 2025 20:42:50
%S A355174 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 A355174 2695,7084,0,1,19,184,1196,5750,20930,53820,0,1,22,247,1872,10647,
%U A355174 47502,166257,420732,0,1,25,319,2755,17980,93496,395560,1344904,3362260
%N A355174 The Fuss-Catalan triangle of order 3, read by rows. Related to quartic trees.
%C A355174 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 A355174 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 A355174 The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
%F A355174 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 A355174 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 A355174 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 A355174 T(n, k) = [x^k] (0^n + (x - 4*x^2)/(1 - x)^(3*n + 2)).
%e A355174 Table T(n, k) begins:
%e A355174   [0] [1]
%e A355174   [1] [0, 1]
%e A355174   [2] [0, 1,  4]
%e A355174   [3] [0, 1,  7,  22]
%e A355174   [4] [0, 1, 10,  49,  140]
%e A355174   [5] [0, 1, 13,  85,  357,   969]
%e A355174   [6] [0, 1, 16, 130,  700,  2695,  7084]
%e A355174   [7] [0, 1, 19, 184, 1196,  5750, 20930,  53820]
%e A355174   [8] [0, 1, 22, 247, 1872, 10647, 47502, 166257,  420732]
%e A355174   [9] [0, 1, 25, 319, 2755, 17980, 93496, 395560, 1344904, 3362260]
%e A355174 Seen as an array reading the diagonals starting from the main diagonal:
%e A355174   [0] 1, 1,  4,  22,  140,   969,   7084,   53820,   420732, ...  A002293
%e A355174   [1] 0, 1,  7,  49,  357,  2695,  20930,  166257,  1344904, ...  A233658
%e A355174   [2] 0, 1, 10,  85,  700,  5750,  47502,  395560,  3321120, ...  A233667
%e A355174   [3] 0, 1, 13, 130, 1196, 10647,  93496,  816816,  7128420, ...
%e A355174   [4] 0, 1, 16, 184, 1872, 17980, 167552, 1535352, 13934752, ...
%o A355174 (Python)
%o A355174 from functools import cache
%o A355174 from itertools import accumulate
%o A355174 @cache
%o A355174 def Trow(n: int) -> list[int]:
%o A355174     if n == 0: return [1]
%o A355174     if n == 1: return [0, 1]
%o A355174     row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
%o A355174     return list(accumulate(accumulate(accumulate(row))))
%o A355174 for n in range(11): print(Trow(n))
%Y A355174 A002293 (main diagonal), A233658 (subdiagonal), A233667 (diagonal 2), A016777 (column 2), A196678 (row sums).
%Y A355174 Cf. A123110 (triangle of order 0), A355173 (triangle of order 1), A355172 (triangle of order 2), A355262 (Fuss-Catalan array).
%K A355174 nonn,tabl
%O A355174 0,6
%A A355174 _Peter Luschny_, Jun 25 2022