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Sum of the zeroth moments of all partitions of n.

Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that any part z is composed of labeled elements of amount 1, i.e., z = 1_1 + 1_2 + ... + 1_z. Then one can take from z a single element in z different ways. E.g., for n=3 to n=2 we have A066186(3) = 9 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [111], [12] --> [111], [12] --> [2], [3] --> 2, [3] --> 2, [3] --> 2. For the unlabeled case, one can take a single element from z in only one way. Then the number of one-element transitions from the integer partitions of n to the partitions of n-1 is given by A000070. E.g., A000070(3) = 4 and for the transition from n=3 to n=2 one has [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. - Thomas Wieder, May 20 2004

Also sum of all parts of all regions of n (Cf. A206437). - Omar E. Pol, Jan 13 2013

From Omar E. Pol, Jan 19 2021: (Start)

Apart from initial zero this is also as follows:

Convolution of A000203 and A000041.

Convolution of A024916 and A002865.

For n >= 1, a(n) is also the number of cells in a symmetric polycube in which the terraces are the symmetric representation of sigma(k), for k = n..1, (cf. A237593) starting from the base and located at the levels A000041(0)..A000041(n-1) respectively. The polycube looks like a symmetric tower (cf. A221529). A dissection is a three-dimensional spiral whose top view is described in A239660. The growth of the volume of the polycube represents each convolution mentioned above. (End)

From Omar E. Pol, Feb 04 2021: (Start)

a(n) is also the sum of all divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned divisors are also all parts of all partitions of n.

Apart from initial zero this is also the convolution of A340793 and A000070. (End)

In other words: in row n replace every term of n-th row of A176206 with its divisors.

The terms in row n are also all parts of all partitions of n.

As in A336812 here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the correspondence between all divisors of all terms of the n-th row of A176206 and all parts of all partitions of n, with n >= 1. Both the mentionded divisors and the mentioned parts are the same numbers (see Example section). That is because all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.

For an equivalent table for all parts of the last section of the set of partitions of n see the subsequence A336812. The section is the smallest substructure of the set of partitions in which appears the correspondence divisor/part.

From Omar E. Pol, Aug 01 2021: (Start)

The terms of row n appears in the triangle A346741 ordered in accordance with the successive sections of the set of partitions of n.

The terms of row n in nonincreasing order give the n-th row of A302246.

The terms of row n in nondecreasing order give the n-th row of A302247.

For the connection with the tower described in A221529 see also A340035. (End)

Here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the corresponce between all parts of the last section of the set of partitions of n and all divisors of all terms of the n-th row of A336811, with n >= 1. The mentionded parts and the mentioned divisors are the same numbers (see Example section).

For an equivalent table showing the same kind of correspondence for all partitions of all positive integers see the supersequence A338156.

For further information about the correspondence divisor/part see A338156.

The sum of row n equals A138879(n), the sum of all parts in the last section of the set of partitions of n.

T(n,k) is also the number of cubic cells (or cubes) added at the n-th stage in the k-th level starting from the base in the tower described in A221529, assuming that the tower is an object under construction (see the example). - Omar E. Pol, Jan 20 2022

For further properties of this triangle see also A182703.

Also due to the correspondence divisor/part row n lists the terms of the n-th row of A338156 in nonincreasing order. In other words: row n lists in nonincreasing order the divisors of the terms of the n-th row of A176206. - Omar E. Pol, Jun 16 2022

Also due to the correspondence divisor/part row n lists the terms of the n-th row of A338156 in nondecreasing order. In other words: row n lists in nondecreasing order the divisors of the terms of the n-th row of A176206. - Omar E. Pol, Jun 16 2022

The tetrahedron shows a connection between divisors and partitions.

The sum of all elements of slice n is A066186(n).

The sum of row j of slice n is A221529(n,j).

The sum of column k of slice n is A138785(n,k), the sum of all parts of size k in all partitions of n.

See also the tetrahedron of A221650.

From Omar E. Pol, Mar 18 2018: (Start)

In the n-th row of the triangle the first differences together with its last term give the n-th row of triangle A138785 (see below):

Row..........: 1 2 3 4 5 ...

--- ---- ------- ------------ ----------------

This triangle: 1; 4, 2; 9, 5, 3; 20, 13, 7, 4; 35, 23, 15, 9, 5; ...

| | /| | /| /| | / | /| /| | / | / | /| /|

| |/ | |/ |/ | |/ |/ |/ | |/ |/ |/ |/ |

A138785......: 1; 2, 2; 4, 2, 3; 7, 6, 3, 4; 12, 8, 6, 4, 5; ... (End)