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A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.

%I A355173 #36 Mar 22 2025 20:42:54
%S A355173 1,0,1,0,1,2,0,1,3,5,0,1,4,9,14,0,1,5,14,28,42,0,1,6,20,48,90,132,0,1,
%T A355173 7,27,75,165,297,429,0,1,8,35,110,275,572,1001,1430,0,1,9,44,154,429,
%U A355173 1001,2002,3432,4862,0,1,10,54,208,637,1638,3640,7072,11934,16796
%N A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.
%C A355173 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 = 1. (See the Python program for a reference implementation.)
%C A355173 This definition also includes the classical Fuss-Catalan numbers, since T(n, n) = A000108(n), or row 2 in A355262. For m = 2 see A355172 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 A355173 The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
%F A355173 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, 1).
%F A355173 T(n, k) = (n - k + 2)*(n + k - 1)!/((n + 1)!*(k - 1)!) for k > 0; T(n, 0) = 0^n.
%F A355173 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 A355173 T(n, k) = [x^k] (0^n + (x - 2*x^2)/(1 - x)^(n + 2)).
%e A355173 Table T(n, k) begins:
%e A355173   [0] [1]
%e A355173   [1] [0, 1]
%e A355173   [2] [0, 1, 2]
%e A355173   [3] [0, 1, 3,  5]
%e A355173   [4] [0, 1, 4,  9,  14]
%e A355173   [5] [0, 1, 5, 14,  28,  42]
%e A355173   [6] [0, 1, 6, 20,  48,  90,  132]
%e A355173   [7] [0, 1, 7, 27,  75, 165,  297, 429]
%e A355173   [8] [0, 1, 8, 35, 110, 275,  572, 1001, 1430]
%e A355173   [9] [0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862]
%e A355173 Seen as an array reading the diagonals starting from the main diagonal:
%e A355173   [0] 1, 1, 2,  5,  14,   42,  132,   429,  1430,   4862,   16796, ...  A000108
%e A355173   [1] 0, 1, 3,  9,  28,   90,  297,  1001,  3432,  11934,   41990, ...  A000245
%e A355173   [2] 0, 1, 4, 14,  48,  165,  572,  2002,  7072,  25194,   90440, ...  A099376
%e A355173   [3] 0, 1, 5, 20,  75,  275, 1001,  3640, 13260,  48450,  177650, ...  A115144
%e A355173   [4] 0, 1, 6, 27, 110,  429, 1638,  6188, 23256,  87210,  326876, ...  A115145
%e A355173   [5] 0, 1, 7, 35, 154,  637, 2548,  9996, 38760, 149226,  572033, ...  A000588
%e A355173   [6] 0, 1, 8, 44, 208,  910, 3808, 15504, 62016, 245157,  961400, ...  A115147
%e A355173   [7] 0, 1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367, 1562275, ...  A115148
%o A355173 (Python)
%o A355173 from functools import cache
%o A355173 from itertools import accumulate
%o A355173 @cache
%o A355173 def Trow(n: int) -> list[int]:
%o A355173     if n == 0: return [1]
%o A355173     if n == 1: return [0, 1]
%o A355173     row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
%o A355173     return list(accumulate(row))
%o A355173 for n in range(11): print(Trow(n))
%Y A355173 A000108 (main diagonal), A000245 (subdiagonal), A002057 (diagonal 2), A000344 (diagonal 3), A000027 (column 2), A000096 (column 3), A071724 (row sums), A000958 (alternating row sums), A262394 (main diagonal of array).
%Y A355173 Variants: A009766 (main variant), A030237, A130020.
%Y A355173 Cf. A123110 (triangle of order 0), A355172 (triangle of order 2), A355174 (triangle of order 3), A355262 (Fuss-Catalan array).
%Y A355173 Cf. A115144, A115145, A000588, A115147, A115148.
%K A355173 nonn,tabl
%O A355173 0,6
%A A355173 _Peter Luschny_, Jun 25 2022