

A283368


Irregular triangle read by rows: T(n,k) = number of heights for the horizontal elements of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259176.


2



1, 2, 3, 2, 4, 3, 5, 3, 6, 5, 4, 7, 5, 4, 8, 6, 5, 9, 7, 5, 10, 8, 7, 6, 11, 8, 7, 6, 12, 10, 9, 7, 13, 10, 9, 7, 14, 11, 9, 8, 15, 12, 11, 10, 8, 16, 13, 12, 11, 9, 17, 13, 12, 11, 9, 18, 15, 13, 12, 10, 19, 15, 13, 12, 10, 20, 16, 15, 13, 11, 21, 17, 16, 15, 14, 11
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OFFSET

1,2


COMMENTS

The dot product of the nth row of this triangle and the nth row of triangle A259176 equals A024916(n), the sum of all divisors of numbers 1 through n (true for all n <= 20000); the value is the sum of the rectangles between the xaxis and the horizontal legs of the symmetric representation of sigma(n). This is the companion computation to A283367.


LINKS

Table of n, a(n) for n=1..76.


FORMULA

T(n,k) = n  sum_{i=1..k1} f(n, 2*i) 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 A259176 and row(n) = floor(sqrt(8*n+1)1)/2).


EXAMPLE

The first horizontal leg of the symmetric representation of sigma(15) is at ycoordinate 15 and has length 8, and row 15 has 5 entries so that T(15,1) = 15 and T(15,5) = 8.
The first 16 rows of the irregular triangle:
1
2
3 2
4 3
5 3
6 5 4
7 5 4
8 6 5
9 7 5
10 8 7 6
11 8 7 6
12 10 9 7
13 10 9 7
14 11 9 8
15 12 11 10 8
16 13 12 11 9


MATHEMATICA

(* function f[n, k] and its support functions are defined in A237593 *)
a283368[n_, k_] := n  Sum[f[n, 2i], {i, k1}]
TableForm[Table[a283368[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
Flatten[Table[a283368[n, k], {n, 1, 21}, {k, 1, row[n]}]] (* sequence data *)


CROSSREFS

Cf. A024916, A237593, A259176, A259177, A283367.
Sequence in context: A157893 A331253 A264116 * A199474 A246694 A054384
Adjacent sequences: A283365 A283366 A283367 * A283369 A283370 A283371


KEYWORD

nonn,tabf


AUTHOR

Hartmut F. W. Hoft, Mar 06 2017


STATUS

approved



