%I #35 Nov 16 2024 02:02:05
%S 1,1,1,1,1,0,0,1,1,1,1,1,1,0,0,1,1,1,2,2,1,1,1,0,0,0,0,1,1,1,1,1,1,1,
%T 1,0,1,1,0,1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0,1,1,1,2,2,2,2,1,1,1,0,0,0,
%U 0,0,0,1,1,1,1,0,0,1,1,1,1,0,1,1,2,2,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1
%N Irregular triangle read by rows in which row n lists the elements of row n of A249223 and then the elements of the same row in reverse order.
%C The n-th row of the triangle has length 2*A003056(n).
%C This sequence extends A249223 in the same manner as A237593 extends A237591.
%C The entries in the n-th row of the triangle are the widths of the regions between the (n-1)-st and n-th Dyck paths for the symmetric representation of sigma(n) with each column representing the corresponding leg of the n-th path.
%H N. J. A. Sloane, <a href="/A262045/b262045.txt">Table of n, a(n) for n = 1..3200</a> [First 150 rows, based on G. C. Greubel's b-file for A249223]
%F T(n, k) = T(n, 2*A003056(n) + 1 - k) = A249223(n, k), for 1 <= n and 1 <= k <= A003056(n).
%e n\k 1 2 3 4 5 6 7 8 9 10
%e 1 1 1
%e 2 1 1
%e 3 1 0 0 1
%e 4 1 1 1 1
%e 5 1 0 0 1
%e 6 1 1 2 2 1 1
%e 7 1 0 0 0 0 1
%e 8 1 1 1 1 1 1
%e 9 1 0 1 1 0 1
%e 10 1 1 1 0 0 1 1 1
%e 11 1 0 0 0 0 0 0 1
%e 12 1 1 2 2 2 2 1 1
%e 13 1 0 0 0 0 0 0 1
%e 14 1 1 1 0 0 1 1 1
%e 15 1 0 1 1 2 2 1 1 0 1
%e 16 1 1 1 1 1 1 1 1 1 1
%e 17 1 0 0 0 0 0 0 0 0 1
%e 18 1 1 2 1 1 1 1 2 1 1
%e 19 1 0 0 0 0 0 0 0 0 1
%e 20 1 1 1 1 2 2 1 1 1 1
%e ...
%e The triangle shows that the region between a Dyck path for n and n-1 has width 1 if n is a power of 2. For n a prime the region is a horizontal rectangle of width (height) 1 and the vertical rectangle of width 1 which is its reflection. The Dyck paths and regions are shown below for n = 1..5 (see the A237593 example for n = 1..28):
%e _ _ _
%e 5 |_ _ _|
%e 4 |_ _ |_ _
%e 3 |_ _|_ | |
%e 2 |_ | | | |
%e 1 |_|_|_|_|_|
%t (* functions a237048[ ] and row[ ] are defined in A237048 *)
%t f[n_] :=Drop[FoldList[Plus, 0, Map[(-1)^(#+1)&, Range[row[n]]] a237048[n]], 1]
%t a262045[n_]:=Join[f[n], Reverse[f[n]]]
%t Flatten[Map[a262045, Range[16]]](* data *)
%Y Cf. A000203, A003056, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A249223, A262048.
%K nonn,tabf,changed
%O 1,19
%A _Hartmut F. W. Hoft_, Sep 09 2015