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All entries in the triangle are nonnegative since the number of 1's in odd-numbered columns of A237048 prior to column j, 1 <= j <= row(n), is at least as large as the number of 1's in even-numbered columns through column j. As a consequence:

(a) The two adjacent symmetric Dyck paths whose legs are defined by adjacent rows of triangle A237593 never cross each other (see also A235791 and A237591) and the rows in this triangle describe the widths between the legs.

(b) Let legs(n) denote the n-th row of triangle A237591, widths(n) the n-th row of this triangle, and c(n) the rightmost entry in the n-th row of this triangle (center of the Dyck path). Then area(n) = 2 * legs(n) . width(n) - c(n), where "." is the inner product, is the area between the two adjacent symmetric Dyck paths.

(c) For certain sequences of integers, it is known that area(n) = sigma(n); see A238443, A245685, A246955, A246956 and A247687.

Right border gives A067742. - Omar E. Pol, Jan 21 2017

For a proof that T(n, k) = |{ d : d|n and k/2 < d <= k }|, for 1 <= k <= row(n), where row(n) is the length of row n, an identity suggested by Peter Munn, see the link. A corollary to it is that the number of divisors of n in the half-open interval (row(n)/2, row(n)] equals the width of the symmetric representation of n at the diagonal: T(n, row(n)) = | { d : d|n and row(n)/2 < d <= row(n) } |. See also the comments and conjectures of Michel Marcus in A067742 and A237593. - Hartmut F. W. Hoft, Jun 24 2024

From Omar E. Pol, Jul 24 2024: (Start)

Conjecture 1: Every column is a periodic sequence.

Conjecture 2: The periods of the columns 1..8 are respectively: 1, 2, 6, 12, 60, 60, 420, 840.

Question 1: Is the period of the column k equal to A003418(k)? (End).

From Omar E. Pol, Jul 26 2024: (Start)

Column 1 gives A000012.

Column 2 gives A000035.

Conjecture 3: Column 3 gives [2, 0] together with A115357, hence column 3 gives 2 together with A171182.

Question 2: Except the first nine terms of A337976, is the column 4 the same as A337976?

Question 3: Except the first 14 terms of A366981, is the column 5 the same as A366981? (End)

From Hartmut F. W. Hoft, Aug 01 2024: (Start)

Conjectures 1 and 2 are true and the answer to question 1 is affirmative.

By definition, each column k in triangle T237048(n, k) of sequence A237048 is a periodic sequence of period k. Since the k-th term in row n of the triangle T(n, k) = Sum_{i=1..k} (-1)^(i+1) * T237048(n, i), with 1 <= k <= A003056(n), each initial subsequence T(n, 1) .. T(n, k) of row n in this triangle is periodic of period lcm(1, .. , k) = A003418(k). This implies that each column k in this sequence has period A003418(k).

Conjecture 3 and Question 2 are true. Since T237048(n, 1) = 1, T237208(n, 2) = 1 if n odd and 0 if n even, T237048(n, 3) = 1 if 3|n and 0 otherwise, and T237048(n, 4) = 1 if 4|(n-2) and 0 otherwise, equations T249223(n, 3) = 1 - (n mod 2) + delta( n mod 3) and T249223(n, 4) = 1 - (n mod 2) + delta( n mod 3) - delta( (n-2) mod 4) hold where delta(k) = 1 if k = 0 and 0 otherwise. With the 3rd column starting at n = A000217(3) = 6, each period starting in a row that is a multiple of 6 is [ 2 0 1 1 1 0 ], and appropriate shifts yield A115357 and A171182. With the 4th column starting at n = A000217(4) = 10, each period starting in a row n with 12|(n+2) is [ 0 0 2 0 0 1 1 0 1 0 1 1 ], and with a shift of 9 yields the apparently periodic A337976(10), A337976(11), ... (End)

Here T(n,k) is defined to be the "k-th width" of the symmetric representation of sigma(n), with n>=1 and 1<=k<=2n-1. Explanation: consider the diagram of the symmetric representation of sigma(n) described in A236104, A237593 and other related sequences. Imagine that the diagram for sigma(n) contains 2n-1 equidistant segments which are parallel to the main diagonal [(0,0),(n,n)] of the quadrant. The segments are located on the diagonal of the cells. The distance between two parallel segment is equal to sqrt(2)/2. T(n,k) is the length of the k-th segment divided by sqrt(2). Note that the triangle contains nonnegative terms because for some n the value of some widths is equal to zero. For an illustration of some widths see Hartmut F. W. Hoft's contribution in the Links section of A237270.

Row n has length 2*n-1.

Row sums give A000203.

If n is a power of 2 then all terms of row n are 1's.

If n is an even perfect number then all terms of row n are 1's except the middle term which is 2.

If n is an odd prime then row n lists (n+1)/2 1's, n-2 zeros, (n+1)/2 1's.

The number of blocks of positive terms in row n gives A237271(n).

The sum of the k-th block of positive terms in row n gives A237270(n,k).

It appears that the middle diagonal is also A067742 (which was conjectured by Michel Marcus in the entry A237593 and checked with two Mathematica functions up to n = 100000 by Hartmut F. W. Hoft).

It appears that the trapezoidal numbers (A165513) are also the numbers k > 1 with the property that some of the noncentral widths of the symmetric representation of sigma(k) are not equal to 1. - Omar E. Pol, Mar 04 2023

For the definition of k-th width of the symmetric representation of sigma(n) see A249351.

Row n list the first n terms of the n-th row of A249351.

It appears that the leading diagonal is also A067742 (which was conjectured by Michel Marcus in the entry A237593 and checked with two Mathematica functions up to n = 100000 by Hartmut F. W. Hoft).

For more information see A237591, A237593.

A number k with two parts in its symmetric representation of sigma(k) [ssrs(k) = 2] has the form k = q*p with q in A174973, p prime and 2*q < p. This implies that 2*q <= row(k) < p and the first 0 in the k-th row of A249223 (having row(k) = floor((sqrt(8*k+1)-1)/2) entries) occurs at position 2*q so that 2*q-1 is the number of legs in each of the two parts. Therefore, the numbers 2*q-1 with q in A174973 are the only possible leg counts when ssrs(k) = 2, and for given q in A174973 and smallest prime p(q) > 2*q the number k = q*p(q) is the smallest with a leg count of 2*q-1. Consequently, each number q*p in the column of the irregular triangle A239929 labeled by q in A174793 with p prime satisfies ssrs(q*p) = 2*q-1.

a(1) = 3 is the only odd number since 1 is the only odd number in A174973.

Every number n = 2^m * p, m >= 0, 2^(m+1) < p and p prime, in this sequence is the sum of 2^(m+1) consecutive positive integers which includes every number in A246956.

Numbers m = 2^k * q, k >= 0 and q > 1 odd, without odd prime factors p < 2^(k+1).

This sequence is a proper subsequence of A238524. Numbers 78 = A370206(1) = A238524(55) and 102 = A237287(72) are not in this sequence since their width pattern (A341969) is 1210121.

A000079 is not a subsequence since SRS(2^k), k>=0, consists of a single part of width 1.

Let m = 2^k * q, k >= 0 and q > 1 odd, be a number in this sequence and s the size of the first part of SRS(m) which has width 1 and consists of 2^(k+1) - 1 legs of width 1. Therefore, s = Sum_{i=1..2^(k+1)-1} a237591(m, i) = a235791(m, 1) - a235791(m, 2^(k+1)) = ceiling((m+1)/1 - (1+1)/2) - ceiling((m+1)/2^(k+1) - (2^(k+1) + 1)/2) = (2^(k+1) - 1)(q+1)/2. In other words, point (m, s) is on the line s(m) = (2^(k+1) - 1)/2^(k+1) * m + (2^(k+1) - 1)/2.

For every odd number m in this sequence, the first part of SRS(m) has size (m+1)/2.

Let u = 2^k * Product_{i=1..PrimePi(2^(k+1)} p_i, where p_i is the i-th prime, and let v be the number of elements in this sequence that are in the set V = {m = 2^k * q | 1 < m <= u } then T(j + t*v, k) = T(j, k) + t*u, 1 <= j and 1 <= t, holds for the elements in column k.

For column k = 1, 2, 3, 4, 5, ... the number of sides of the mentioned s-gon are respectively 4, 8, 16, 32, 64, ...

Conjecture 1: column k gives the row numbers of the triangle A364639 where the rows are [1, A036563(k+1) zeros, -1, 1] or where the rows start with [1, A036563(k+1) zeros, -1, 1] and the remaining terms are zeros.

Conjecture 2: every column gives a subsequence of A246955.

Conjecture 3: the sequence is infinite.

Observation 1: at least the terms <= 199 in increasing order coincide with at least the first 82 terms of the intersection of A071561 and A365406.

Observation 2: in the Example section of A246955 there is an irregular triangle. It seems that the terms sorted of the triangle give the sequence A246955. At least the first r(k) terms in the column (k - 1) of the triangle coincide with the first r(k) terms of the column k of this square array, where r(k) are 19, 18, 16, 14, 7 for k = 1..5 respectively.

Observation 3: at least the first five terms of the row 1 coincide with the first five terms of A246956.