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Row n lists in nonincreasing order the first A014153(n-1) terms of A176206.

In other words: row n lists in nonincreasing order the terms of the first n rows of triangle A176206.

Conjecture: all divisors of all terms in row n are also all parts of all partitions of all positive integers <= n.

The conjecture is in accordance with the conjectures in A336811 and in A176206.

A336811 contains the most elementary conjecture about the correspondence divisors/partitions.

The connection with A336811 (the main sequence) is as follows: A336811 --> A176206 --> this sequence.

Also the total number of all different integers in all partitions of n + 1. E.g., a(3) = 7 because the partitions of 4 comprise the sets {1},{1, 2},{2},{1, 3},{4} of different integers and their total number is 7. - Thomas Wieder, Apr 10 2004

With offset 1, also the number of 1's in all partitions of n. For example, 3 = 2+1 = 1+1+1, a(3) = (zero 1's) + (one 1's) + (three 1's), so a(3) = 4. - Naohiro Nomoto, Jan 09 2002. See the Riordan reference p. 184, last formula, first term, for a proof based on Fine's identity given in Riordan, p. 182 (20).

Also, number of partitions of n into parts when there are two kinds of parts of size one.

Also number of graphical forest partitions of 2n+2.

a(n) = count 2 for each partition of n and 1 for each decrement. E.g., the partitions of 4 are 4 (2), 31 (3), 22 (2), 211 (3) and 1111 (2). 2 + 3 + 2 + 3 + 2 = 12. This is related to the Ferrers representation. We can see that taking the Ferrers diagram for each partition of n and adding a new * to all available columns, we generate each partition of n+1, but with repeats (A058884). - Jon Perry, Feb 06 2004

Also the number of 1-transitions among all integer partitions of n. A 1-transition is the removal of a digit "1" from a partition containing at least one "1" and subsequent addition of that "1" to another digit in that partition. This other digit may be a "1" also, but all digits of equal amount are considered as undistinquishable (unlabeled). E.g., for n=6 one has the partition [1113] for which the following two 1-transitions are possible: [1113] --> [123] and [1113] --> [114]. The 1-transitions of n form a partial order (poset). For n=6 one has 12 1-transitions: [111111] --> [11112], [11112] --> [1113], [11112] --> [1122], [1113] --> [114], [1113] --> [123], [1122] --> [123], [1122] --> [222], [123] --> [33], [123] --> [24], [114] --> [15], [114] --> [24], [15] --> [6]. - Thomas Wieder, Mar 08 2005

Also number of partitions of 2n+1 where one of the parts is greater than n (also where there are more than n parts) and of 2n+2 where one of the parts is greater than n+1 (or with more than n+1 parts). - Henry Bottomley, Aug 01 2005

Equals left border of triangle A137633 - Gary W. Adamson, Jan 31 2008

Equals row sums of triangle A027293. - Gary W. Adamson, Oct 26 2008

Convolved with A010815 = [1,1,1,...]. n-th partial sum of A000041 convolved with A010815 = the binomial sequence starting (1, n, ...). - Gary W. Adamson, Nov 09 2008

Equals A036469 convolved with A035363. - Gary W. Adamson, Jun 09 2009

a(A004526(n)) = A025065(n). - Reinhard Zumkeller, Jan 23 2010

a(n) = if n <= 1 then A054225(1,n) else A054225(n,1). - Reinhard Zumkeller, Nov 30 2011

Also the total number of 1's among all hook-lengths in all partitions of n. E.g., a(4)=7 because hooks of the partitions of n = 4 comprise the multisets {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1} and their total number of 1's is 7. - T. Amdeberhan, Jun 03 2012

With offset 1, a(n) is also the difference between the sum of largest and the sum of second largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

a(0) = 1 and 2*a(n-1) >= a(n) for all n > 0. Hence a(n) is a complete sequence. - Frank M Jackson, Apr 08 2013

a(n) is the number of conjugacy classes in the order-preserving, order-decreasing and (order-preserving and order-decreasing) injective transformation semigroups. - Ugbene Ifeanyichukwu, Jun 03 2015

a(n) is also the number of unlabeled subgraphs of the n-cycle C_n. For example, for n = 3, there are 3 unlabeled subgraphs of the triangle C_3 with 0 edges, 2 with 1 edge, 1 with 2 edges, and 1 with 3 edges (C_3 itself), so a(3) = 3 + 2 + 1 + 1 = 7. - John P. McSorley, Nov 21 2016

a(n) is also the number of partitions of 2n with all parts either even or equal to 1. Proof: the number of such partitions of 2n with exactly 2k 1's is p(n-k), for k = 0,..,n. Summing over k gives the formula. - Leonard Chastkofsky, Jul 24 2018

a(n) is the total number of polygamma functions that appear in the expansion of the (n+1)st derivative of x! with respect to x. More specifically, a(n) is the number of times the string "PolyGamma" appears in the expansion of D[x!, {x, n + 1}] in Mathematica. For example, D[x!, {x, 3 + 1}] = Gamma[1 + x] PolyGamma[0, 1 + x]^4 + 6 Gamma[1 + x] PolyGamma[0, 1 + x]^2 PolyGamma[1, 1 + x] + 3 Gamma[1 + x] PolyGamma[1, 1 + x]^2 + 4 Gamma[1 + x] PolyGamma[0, 1 + x] PolyGamma[2, 1 + x] + Gamma[1 + x] PolyGamma[3, 1 + x], and we see that the string "PolyGamma" appears a total of a(3) = 7 times in this expansion. - John M. Campbell, Aug 11 2018

With offset 1, also the number of integer partitions of 2n that do not comprise the multiset of vertex-degrees of any multigraph (i.e., non-multigraphical partitions); see A209816 for multigraphical partitions. - Gus Wiseman, Oct 26 2018

Also a(n) is the number of partitions of 2n+1 with exactly one odd part.

Delete the odd part 2k+1, k=0, ..., n, to get a partition of 2n-2k into even parts. There are as many unrestricted partitions of n-k; now sum those numbers from 0 to n to get a(n). - George Beck, Jul 22 2019

In the Young's lattice, a(n) is the number of branches that connect the (n-1)-th layer to the n-th layer. - Shouvik Datta, Sep 19 2021

a(n) is the number of multiset partitions of the multiset {r^n, s^1}, equivalently, factorization patterns of any number m=p^n*q^1 where p and q are primes. - Joerg Arndt, Jan 01 2024

a(n) is the number of positive integers whose divisors are the parts of the partitions of n + 1. - Omar E. Pol, Nov 07 2024

Conjecture: Reverse the rows of the table to get an infinite lower-triangular matrix b with 1's on the main diagonal. The third diagonal of the inverse of b is minus A137719. - George Beck, Oct 26 2019

Proof: The reversed rows yield the matrix I+N where N is strictly lower triangular, N[i,j] = 0 for j >= i, having its 2nd diagonal equal to the 2nd column (1, 0, 1, 0, 1, ...): N[i+1,i] = A000035(i), i >= 1, and 3rd diagonal equal to the 3rd column of this triangle, (2, 1, 2, 3, 3, 3, ...): N[i+2,i] = A137719(i), i >= 1. It is known that (I+N)^-1 = 1 - N + N^2 - N^3 +- .... Here N^2 has not only the second but also the 3rd diagonal zero, because N^2[i+2,i] = N[i+2,i+1]*N[i+1,i] = A000035(i+1)*A000035(i) = 0. Therefore the 3rd diagonal of (I+N)^-1 is equal to -A137719 without leading 0. - M. F. Hasler, Oct 27 2019

From Gus Wiseman, Aug 27 2023: (Start)

Also the number of ways to write n-k as a nonnegative linear combination of a strict integer partition of k. Also the number of ways to write n as a (strictly) positive linear combination of a strict integer partition of k. Row n=7 counts the following:

7*1 . 1*2+5*1 1*3+4*1 1*3+2*2 1*5+2*1 1*7

2*2+3*1 2*3+1*1 1*4+3*1 1*3+1*2+2*1 1*4+1*3

3*2+1*1 1*5+1*2

1*6+1*1

1*4+1*2+1*1

(End)

Also number of partitions of 2*n with exactly 2 odd parts (offset 1). - Vladeta Jovovic, Jan 12 2005

Also number of transitions from one partition of n+2 to another, where a transition consists of replacing any two parts with their sum. Remove all 1' and 2' from the partition, replacing them with ((number of 2') + 1) and ((number of 1') + (number of 2') + 1); these are the two parts being summed. Number of partitions of n into parts of 2 kinds with at most 2 parts of the second kind, or of n+2 into parts of 2 kinds with exactly 2 parts of the second kind. - Franklin T. Adams-Watters, Mar 20 2006

From Christian Gutschwager (gutschwager(AT)math.uni-hannover.de), Feb 10 2010: (Start)

a(n) is also the number of pairs of partitions of n+2 which differ by only one box (for bijection see the first Gutschwager link).

a(n) is also the number of partitions of n with two parts marked.

a(n) is also the number of partitions of n+1 with two different parts marked. (End)

Convolution of A000041 and A008619. - Vaclav Kotesovec, Aug 18 2015

a(n) = P(/2,n), a particular case of P(/k,n) defined as follows: P(/0,n) = A000041(n) and P(/k,n) = P(/k-1, n) + P(/k-1,n-k) + P(/k-1, n-2k) + ... Also, P(/k,n) = the convolution of A000041 and the partitions of n with exactly k parts, and g.f. P(/k,n) = (g.f. for P(n)) * 1/(1-x)...(1-x^k). - Gregory L. Simay, Mar 22 2018

a(n) is also the sum of binomial(D(p),2) in partitions p of (n+3), where D(p)= number of different sizes of parts in p. - Emily Anible, Apr 03 2018

Also partitions of 2*(n+1) with alternating sum 2. Also partitions of 2*(n+1) with reverse-alternating sum -2 or 2. - Gus Wiseman, Jun 21 2021

Define the distance graph of the partitions of n using the distance function in A366156 as follows: two vertices (partitions) share an edge if and only if the distance between the vertices is 2. Then a(n) is the number of edges in the distance graph of the partitions of n. - Clark Kimberling, Oct 12 2023

The original definition was: An irregular table: Row n begins with n, counts down to 1 and repeats the intermediate numbers as often as given by the partition numbers.

Row n contains a decreasing sequence where n-k is repeated A000041(k) times, k = 0..n-1.

From Omar E. Pol, Nov 23 2020: (Start)

Row n lists in nonincreasing order the first A000070(n-1) terms of A336811.

In other words: row n lists in nonincreasing order the terms from the first n rows of triangle A336811.

Conjecture: all divisors of all terms in row n are also all parts of all partitions of n.

For more information see the example and A336811 which contains the most elementary conjecture about the correspondence divisors/partitions.

Row sums give A014153.

A338156 lists the divisors of every term of this sequence.

The n-th row of A340581 lists in nonincreasing order the terms of the first n rows of this triangle.

For a regular triangle with the same row sums see A141157. (End)

From Omar E. Pol, Jul 31 2021: (Start)

The number of k's in row n is equal to A000041(n-k), 1 <= k <= n.

The number of terms >= k in row n is equal to A000070(n-k), 1 <= k <= n.

The number of k's in the first n rows (or in the first A014153(n-1) terms of the sequence) is equal to A000070(n-k), 1 <= k <= n.

The number of terms >= k in the first n rows (or in the first A014153(n-1) terms of the sequence) is equal to A014153(n-k), 1 <= k <= n. (End)

Partial sum operator applied to column k gives column k+1.

A(n,k) is also defined for k < 0. All given formulas and programs can be applied also if k is negative.

Conjecture: all divisors of all terms of row n are also all parts of all partitions of n.

The conjecture gives a correspondence between divisors and partitions (see example).

It is conjectured that every section of the set of partitions of n has essentially the same correspondence. For more information see A336811.

Essentially a duplicate of A053222.

Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.

Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.

Convolved with A000027 gives A024916.

Convolved with A000041 gives A138879.

Convolved with A000070 gives the nonzero terms of A066186.

Convolved with the nonzero terms of A002088 gives A086733.

Convolved with A014153 gives A182738.

Convolved with A024916 gives A000385.

Convolved with A036469 gives the nonzero terms of A277029.

Convolved with A091360 gives A276432.

Convolved with A143128 gives the nonzero terms of A000441.

For the correspondence between divisors and partitions see A336811.

a(n) is also the volume of a three-dimensional version of the section model of partitions: the 3D illustrations in A135010 show boxes with face areas of 1 X 1, 2 X 2, 3 X 3, 4 X 5, 5 X 7 units along the m and p(m) axis, which is sequence A066186. Assuming that the boxes are 1 unit deep, the total volume of all boxes up to layer n is a(n). See the first two links.

From Omar E. Pol, Jan 20 2021: (Start)

a(n) is the sum of all parts of all partitions of all positive integers <= n.

Convolution of A000203 and A000070.

Convolution of A024916 and A000041.

Convolution of A175254 and A002865.

Convolution of A340793 and A014153.

Row sums of triangles A340527, A340531, A340579.

Consider a symmetric tower (a polycube) in which the terraces are the symmetric representation of sigma (n..1) respectively starting from the base (cf. A237270, A237593). The total area of the terraces equals A024916(n), the same as the area of the base.

The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1), hence the differences between two successive levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).

a(n) is the volume (or the total number of unit cubes) of the polycube.

That is due to the correspondence between divisors and partitions (cf. A336811).

The symmetric tower is a member of the family of the pyramid described in A245092.

The growth of the volume of the polycube represents every convolution mentioned above. (End)

For k > 0, column k is asymptotic to sqrt(3) * (2*k+1) * exp(Pi*sqrt(2*n/3)) / (2 * k^2 * (k+1)^2 * Pi^2) ~ 6 * (2*k+1) * n * p(n) / (k^2 * (k+1)^2 * Pi^2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, May 29 2018