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A283367
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Irregular triangle read by rows: T(n,k) = number of horizontal positions for the vertical legs of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259177.
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2
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1, 2, 2, 3, 3, 4, 3, 5, 4, 5, 6, 4, 5, 7, 5, 6, 8, 5, 7, 9, 6, 7, 8, 10, 6, 7, 8, 11, 7, 9, 10, 12, 7, 9, 10, 13, 8, 9, 11, 14, 8, 10, 11, 12, 15, 9, 11, 12, 13, 16, 9, 11, 12, 13, 17, 10, 12, 13, 15, 18, 10, 12, 13, 15, 19, 11, 13, 15, 16, 20, 11, 14, 15, 16, 17, 21
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OFFSET
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1,2
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COMMENTS
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The dot product of the n-th row of this triangle and the n-th row of triangle A259177 equals A024916(n), the sum of all divisors of numbers 1 through n (true for all n <= 20000); the value of a(n) is the sum of the rectangles between the y-axis and the vertical legs of the symmetric representation of sigma(n). This is the companion computation to A283368.
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LINKS
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FORMULA
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T(n,k) = sum_{i=1..k} f(n, 2*i-1) where f is defined in A237593.
A024916(n) = sum_{i=1..row(n)} (T(n,i))*S(n,i)) where S(n,i) refers to the triangle of A259177 and row(n) = floor(sqrt(8*n+1)-1)/2).
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EXAMPLE
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The first vertical leg of the symmetric representation of sigma(15) is at x-coordinate 8 and has length 3, and row 15 has 5 entries so that T(15,1) = 8 and T(15,5) = 15.
The first 16 rows of the irregular triangle:
1: 1
2: 2
3: 2 3
4: 3 4
5: 3 5
6: 4 5 6
7: 4 5 7
8: 5 6 8
9: 5 7 9
10: 6 7 8 10
11: 6 7 8 11
12: 7 9 10 12
13: 7 9 10 13
14: 8 9 11 14
15: 8 10 11 12 15
16: 9 11 12 13 16
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MATHEMATICA
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(* function f[n, k] and its support functions are defined in A237593 *)
a283367[n_, k_] := Sum[f[n, 2*i-1], {i, k}]
TableForm[Table[a283367[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
Flatten[Table[a283367[n, k], {n, 1, 21}, {k, 1, row[n]}]] (* sequence data *)
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CROSSREFS
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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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