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A262045
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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.
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31
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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, 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, 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
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OFFSET
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1,19
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COMMENTS
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The n-th row of the triangle has length 2*A003056(n).
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.
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LINKS
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FORMULA
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EXAMPLE
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n\k 1 2 3 4 5 6 7 8 9 10
1 1 1
2 1 1
3 1 0 0 1
4 1 1 1 1
5 1 0 0 1
6 1 1 2 2 1 1
7 1 0 0 0 0 1
8 1 1 1 1 1 1
9 1 0 1 1 0 1
10 1 1 1 0 0 1 1 1
11 1 0 0 0 0 0 0 1
12 1 1 2 2 2 2 1 1
13 1 0 0 0 0 0 0 1
14 1 1 1 0 0 1 1 1
15 1 0 1 1 2 2 1 1 0 1
16 1 1 1 1 1 1 1 1 1 1
17 1 0 0 0 0 0 0 0 0 1
18 1 1 2 1 1 1 1 2 1 1
19 1 0 0 0 0 0 0 0 0 1
20 1 1 1 1 2 2 1 1 1 1
...
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):
_ _ _
5 |_ _ _|
4 |_ _ |_ _
3 |_ _|_ | |
2 |_ | | | |
1 |_|_|_|_|_|
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MATHEMATICA
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(* functions a237048[ ] and row[ ] are defined in A237048 *)
f[n_] :=Drop[FoldList[Plus, 0, Map[(-1)^(#+1)&, Range[row[n]]] a237048[n]], 1]
a262045[n_]:=Join[f[n], Reverse[f[n]]]
Flatten[Map[a262045, Range[16]]](* data *)
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CROSSREFS
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Cf. A000203, A003056, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A249223, A262048.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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