cp's OEIS Frontend

The first array is described in A126442 and is the hook case. Sequence A129305 encodes the multisets counted by A096443 and A129306.

Multinomials count permutations of multisets and also paths in lattices; for example, there are six paths (from null to full) through the lattice of divisors for signature 36: 2233 2323 2332 3223 3232 and 3322.

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

Number of partitions of n with three kinds of 1. E.g., a(2)=7 because we have 2, 1+1, 1+1', 1+1", 1'+1', 1'+1", 1"+1". - Emeric Deutsch, Mar 22 2005

Partial sums of the partial sums of the partition numbers A000041. Partial sums of A000070. Euler transform of 3,1,1,1,...

Also sum of parts, counted without multiplicity, in all partitions of n, offset 1. Also Sum phi(p), where the sum is taken over all parts p of all partitions of n, offset 1. - Vladeta Jovovic, Mar 26 2005

Equals row sums of triangle A141157. - Gary W. Adamson, Jun 12 2008

A014153 convolved with A010815 = (1, 2, 3, ...). n-th partial sum sequence of A000041 convolved with A010815 = (n-1)-th column of Pascal's triangle, starting (1, n, ...). - Gary W. Adamson, Nov 09 2008

From Omar E. Pol, May 25 2012: (Start)

a(n) is also the sum of all parts of the (n+1)st column of a version of the section model of partitions in which every section has its parts aligned to the right margin (cf. A210953, A210970, A135010).

Rows of triangle A210952 converge to this sequence. (End)

Using the above result (see Jovovic's comment) of Jovovic and Mertens's theorem on the average order of the phi function, we can obtain the estimate a(n-1) = (6/Pi^2)*n*p(n) + O(log(n)*A006128(n)), where p(n) is the partition function A000041(n). It can be shown that A006128(n) = O(sqrt(n)*log(n)*p(n)), so we have the asymptotic result a(n) ~ (6/Pi^2)*n*p(n). - Peter Bala, Dec 23 2013

a(n-2) is the number of partitions of 2n or 2n-1 with palindromicity 2; that is, partitions that can be listed in palindromic order except for a central sequence of two distinct parts. - Gregory L. Simay, Nov 01 2015

Convolution of A000041 and A000027. - Omar E. Pol, Jun 17 2021

Convolution of A002865 and the positive terms of A000217. Partial sums give A014160. - Omar E. Pol, Mar 01 2023

Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1); A(3,1) = 7: 2*2*2*3, 2*3*4, 4*6, 2*2*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.

Contribution from George Beck, Jan 08 2011: (Start)

The number of multiset partitions of the n-multiset M={0,0,...,0,1,2} (with n-2 zeros) is sum_{k=0..(n-2)}( (n-k) * p(k) ) where p(k) is the number of partitions of k.

Proof:

For each k = 0, 1, ..., n-2, partition k zeros and add the remaining n-k-2 zeros to the block {1, 2}, to give p(k) partitions.

For each k, partition k zeros and add the remaining n-k-2 zeros to the two blocks {1} and {2} in all possible 1 + n-k-2 ways, which gives (1 + n-k-2) * p(k) partitions.

Together, the number of partitions of M is sum_{k=0..n-2}( (n-k) * p(k) ). (End)

A082775 is the special case of A126442 with n-k = 2.

This triangle shows the same numbers in each row as A129306 and A096443, but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.

Row lengths: A000041 (partition numbers), Row sums: A035310

Columns: 0: A000110 (Bell), 1: A035098 (near-Bell), 2: A169587, 4: A169588

Last in row: end-1: A091437, end: A000041 (partition numbers)

The rightmost columns form a reflected version of the triangle A126442:

n 0 1 2 4 6 10 14 21 (A000041(1,2,3...)-1)

m

1 1

2 2 2

3 5 4 3

4 15 11 7 5

5 52 36 21 12 7

6 203 135 74 38 19 11

7 877 566 296 141 64 30 15

8 4140 2610 1315 592 250 105 45 22

A249619 shows the number of permutations of the same multisets.