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A355172 The Fuss-Catalan triangle of order 2, read by rows. Related to ternary trees.

%I A355172 #28 Mar 22 2025 20:43:00
%S A355172 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,
%T A355172 1428,0,1,13,88,408,1428,3876,7752,0,1,15,117,627,2565,8379,21945,
%U A355172 43263,0,1,17,150,910,4235,15939,49588,126500,246675
%N A355172 The Fuss-Catalan triangle of order 2, read by rows. Related to ternary trees.
%C A355172 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.)
%C A355172 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.
%F A355172 The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
%F A355172 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).
%F A355172 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.
%F A355172 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 A355172 T(n, k) = [x^k] (0^n + (x - 3*x^2)/(1 - x)^(2*n + 2)).
%e A355172 Table T(n, k) begins:
%e A355172   [0] [1]
%e A355172   [1] [0, 1]
%e A355172   [2] [0, 1,  3]
%e A355172   [3] [0, 1,  5, 12]
%e A355172   [4] [0, 1,  7, 25,  55]
%e A355172   [5] [0, 1,  9, 42, 130,  273]
%e A355172   [6] [0, 1, 11, 63, 245,  700, 1428]
%e A355172   [7] [0, 1, 13, 88, 408, 1428, 3876, 7752]
%e A355172 Seen as an array reading the diagonals starting from the main diagonal:
%e A355172   [0] 1, 1,  3, 12,  55,  273,  1428,   7752,   43263,  246675, ...  A001764
%e A355172   [1] 0, 1,  5, 25, 130,  700,  3876,  21945,  126500,  740025, ...  A102893
%e A355172   [2] 0, 1,  7, 42, 245, 1428,  8379,  49588,  296010, 1781325, ...  A102594
%e A355172   [3] 0, 1,  9, 63, 408, 2565, 15939,  98670,  610740, 3786588, ...  A230547
%e A355172   [4] 0, 1, 11, 88, 627, 4235, 27830, 180180, 1157013, 7396972, ...
%o A355172 (Python)
%o A355172 from functools import cache
%o A355172 from itertools import accumulate
%o A355172 @cache
%o A355172 def Trow(n: int) -> list[int]:
%o A355172     if n == 0: return [1]
%o A355172     if n == 1: return [0, 1]
%o A355172     row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
%o A355172     return list(accumulate(accumulate(row)))
%o A355172 for n in range(9): print(Trow(n))
%Y A355172 A001764 (main diagonal), A102893 (subdiagonal), A102594 (diagonal 2), A230547 (diagonal 3), A005408 (column 2), A071355 (column 3), A006629 (row sums), A143603 (variant).
%Y A355172 Cf. A123110 (triangle of order 0), A355173 (triangle of order 1), A355174 (triangle of order 3), A355262 (Fuss-Catalan array).
%K A355172 nonn,tabl
%O A355172 0,6
%A A355172 _Peter Luschny_, Jun 25 2022