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Since A000203(k) has a symmetric representation, both T(n,k) and the partial sums of row n can be represented by symmetric polycubes. For more information see A237593 and A237270. For another version see A245099. - Omar E. Pol, Jul 15 2014

From Omar E. Pol, Jul 10 2021: (Start)

The above comment refers to a symmetric tower whose terraces are the symmetric representation of sigma(i), for i = 1..n, starting from the top. The levels of these terraces are the partition numbers A000041(h-1), for h = 1 to n, starting from the base of the tower, where n is the length of the largest side of the base.

The base of the tower is the symmetric representation of A024916(n).

The height of the tower is equal to A000041(n-1).

The surface area of the tower is equal to A345023(n).

The volume (or the number of cubes) of the tower equals A066186(n).

The volume represents the n-th term of the convolution of A000203 and A000041, that is A066186(n).

Note that the terraces that are the symmetric representation of sigma(n) and the terraces that are the symmetric representation of sigma(n-1) both are unified in level 1 of the structure. That is because the first two partition numbers A000041 are [1, 1].

The tower is an object of the family of the stepped pyramid described in A245092.

T(n,k) can be represented with a set of A237271(k) right prisms of height A000041(n-k) since T(n,k) is the total number of cubes that are exactly below the parts of the symmetric representation of sigma(k) in the tower.

T(n,k) is also the sum of all divisors of all k's that are in the first n rows of triangle A336811, or in other words, in the first A000070(n-1) terms of the sequence A336811. Hence T(n,k) is also the sum of all divisors of all k's in the n-th row of triangle A176206.

The mentioned property is due to the correspondence between divisors and parts explained in A338156: all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.

Therefore the set of all partitions of n >= 1 has an associated tower.

The partial column sums of A340583 give this triangle showing the growth of the structure of the tower.

Note that the convolution of A000203 with any integer sequence S can be represented with a symmetric tower or structure of the same family where its terraces are the symmetric representation of sigma starting from the top and the heights of the terraces starting from the base are the terms of the sequence S. (End)

Row sums = A000041 starting with offset 1.

Sum of n-th row terms = leftmost term of next row, such that terms in each row demonstrate Euler's pentagonal theorem.

Let Q = triangle A027293 with partition numbers in each column.

Let M = a diagonalized variant of A080995 as the characteristic function of the generalized pentagonal numbers starting with offset 1: (1, 1, 0, 0, 1,...)

Sign the 1's: (++--++...) getting (1, 1, 0, 0, -1, 0, -1,...) which is the diagonal of matrix M, (as an infinite lower triangular matrix with the rest zeros).

Triangle A175003 = Q*M, with deleted zeros.

Column k starts at row A001318(k). - Omar E. Pol, Sep 21 2011

From Omar E. Pol, Apr 22 2014: (Start)

Row n has length A235963(n).

For Euler's pentagonal theorem for the sum of divisors see A238442.

Note that both of Euler's pentagonal theorems refer to generalized pentagonal numbers (A001318), not to pentagonal numbers (A000326). (End)

T(n,k) is the number of partitions of n that contain k as a part multiplied by the number of divisors of k.

It appears that T(n,k) is also the total number of appearances of k in the last k sections of the set of partitions of n multiplied by the number of divisors of k.

T(n,k) is also the number of partitions of k into equal parts multiplied by the number of ones in the j-th section of the set of partitions of n, where j = (n - k + 1).

For another version see A245095. - Omar E. Pol, Jul 15 2014

Number of partitions of n into distinct parts with maximal size, see A000009. - Reinhard Zumkeller, Jun 13 2009

Equals A092080 when first column is removed. Georg Fischer, Jul 26 2023

T(n,k) is the sum of the parts of all partitions of n that contain k as a part, assuming that all partitions of n have 0 as a part: Thus, column 0 gives the sum of the parts of all partitions of n.

By definition all entries in row n>0 are divisible by n.

Row sums are 0, 2, 8, 21, 48, 95, 180, 315, 536, 873, 1390, 2145,...

The partitions of n+k that contain k as a part can be obtained by adding k to every partition of n assuming that all partitions of n have 0 as a part.

For example, the partitions of 6+k that contain k as a part are

k + 6

k + 3 + 3

k + 4 + 2

k + 2 + 2 + 2

k + 5 + 1

k + 3 + 2 + 1

k + 4 + 1 + 1

k + 2 + 2 + 1 + 1

k + 3 + 1 + 1 + 1

k + 2 + 1 + 1 + 1 + 1

k + 1 + 1 + 1 + 1 + 1 + 1

The partition number A000041(n) is also the number of partitions of m*(n+k) into parts divisible by m and that contain m*k as a part, with k>=0, m>=1, n>=0 and assuming that all partitions of n have 0 as a part.

By definition, the entries in row n are divisible by n.

Row sums are 1, 4, 12, 28, 60, 114, ... = n*A000070(n).

Column 1 is A228816. - Omar E. Pol, Sep 25 2013

Reluctant sequence of A000203.

Row sums give A024916.

Has a symmetric representation - for more information see A237270.

Row sums yield A015723. T(n,1)=A025147(n-1); T(n,2)=A015744(n-2); T(n,3)=A015745(n-3); T(n,4)=A015746(n-4); T(n,5)=A015750(n-5). - Emeric Deutsch, Mar 29 2006

Number of parts of size k in all partitions of n into distinct parts. Number of partitions of n-k into distinct parts not including a part of size k. - Franklin T. Adams-Watters, Jan 24 2012

This infinite lower triangular Riordan matrix T is the so-called L-eigen-matrix of the infinite lower triangular Riordan matrix A027293 (but with offset 0 for rows and columns). Such eigentriangles have been considered by Paul Barry in the paper given as a link in A186020.

This means that E is the L-eigen-matrix of an infinite lower triangular matrix M if M*E = L*(E - I), with the unit matrix I and the matrix L with elements L(i, j) = delta_{i, j-1} (Kronecker's delta-symbol; first upper diagonal with 1's).

Therefore, this notion is analogous to calling sequence S an L-eigen-sequence of matrix M if M*vec(S) = L.vec(S) (or vec(S) is an eigensequence of M - L with eigenvalue 0), used by Bernstein and Sloane, see the links in A155002.

L*(E - I) is the E matrix after elimination of the main diagonal and then the first row, and starting with offset 0. Because for infinite lower triangular matrices L^{tr}.L = I (tr stands for transposed), this leads to M = L*(I - E^{-1}) or E = (I - L^{tr}*M)^{-1}.

Note that Gary W. Adamson uses a different notion: E is the eigentriangle of a triangle T if the columns of E are the columns j of T multiplied by the sequence elements B_j of B with o.g.f. x/(1 - x*G(x)), with the o.g.f. G(x) of column no. 1 of T. Or E(i, j) = T(i, j)*B(j). In short: sequence B is the L-eigen-sequence of the infinite lower triangular matrix T (but with offset 1): T*vec(B) = L.vec(B). See, e.g., A143866.

Thanks to Gary W. Adamson for motivating my occupation with such eigentriangles and eigensequences.

The first column of the present triangle T is A067687, which is then shifted downwards (Riordan of Toeplitz type).