

A283367


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.


2



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

1,2


COMMENTS

The dot product of the nth row of this triangle and the nth 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 yaxis and the vertical legs of the symmetric representation of sigma(n). This is the companion computation to A283368.


LINKS



FORMULA

T(n,k) = sum_{i=1..k} f(n, 2*i1) 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).


EXAMPLE

The first vertical leg of the symmetric representation of sigma(15) is at xcoordinate 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


MATHEMATICA

(* function f[n, k] and its support functions are defined in A237593 *)
a283367[n_, k_] := Sum[f[n, 2*i1], {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 *)


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



