cp's OEIS Frontend

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

a(0) = 1 because the empty partition vacuously has each part >= 3. - Jason Kimberley, Jan 11 2011

Number of partitions where the largest part occurs at least three times. - Joerg Arndt, Apr 17 2011

By removing a single part of size 3, an A026796 partition of n becomes an A008483 partition of n - 3.

For n >= 3 the sequence counts the isomorphism classes of authentication codes AC(2,n,n) with perfect secrecy and with largest probability 0.5 that an interceptor could deceive with a substituted message. - E. Keith Lloyd (ekl(AT)soton.ac.uk).

For n >= 1, also the number of regular graphs of degree 2. - Mitch Harris, Jun 22 2005

(1 + 0*x + 0*x^2 + x^3 + x^4 + x^5 + 2*x^6 + ...) = (1 + x + 2*x^2 + 3*x^3 + 5*x^4 + ...) * 1 / (1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + ...). - Gary W. Adamson, Jun 30 2009

Because the triangle A051031 is symmetric, a(n) is also the number of (n-3)-regular graphs on n vertices. Since the disconnected (n-3)-regular graph with minimum order is 2K_{n-2}, then for n > 4 there are no disconnected (n-3)-regular graphs on n vertices. Therefore for n > 4, a(n) is also the number of connected (n-3)-regular graphs on n vertices. - Jason Kimberley, Oct 05 2009

Number of partitions of n+2 such that 2*(number of parts) is a part. - Clark Kimberling, Feb 27 2014

For n >= 1, a(n) is the number of (1,1)-separable partitions of n, as defined at A239482. For example, the (1,1)-separable partitions of 11 are [10,1], [7,1,2,1], [6,1,3,1], [5,1,4,1], [4,1,2,1,2,1], [3,1,3,1,2,1], so that a(11) = 6. - Clark Kimberling, Mar 21 2014

From Peter Bala, Dec 01 2024: (Start)

Let P(3, n) denote the set of partitions of n into parts k >= 3. Then A000041(n) = (1/2) * Sum_{parts k in all partitions in P(3, n+3)} phi(k), where phi(k) is the Euler totient function (see A000010). For example, with n = 5, there are 3 partitions of n + 3 = 8 into parts greater then 3, namely, 8, 5 + 3 and 4 + 4, and (1/2)*(phi(8) + phi(5) + phi(3) + 2*phi(4)) = 7 = A000041(5). (End)

Or, number of partitions of n in which number of largest parts is equal to the largest part.

a(n) is the number of partitions of n-1 without parts that differ by less than 2 and which have no parts less than three. [MacMahon]

There are two conflicting choices for the offset in this sequence. For the definition given here the offset is 1, and that is what we shall adopt. On the other hand, if one arrives at this sequence via the Rogers-Ramanujan identities (see the next comment), the natural offset is 0.

Related to Rogers-Ramanujan identities: Let G[1](q) and G[2](q) be the generating functions for the two Rogers-Ramanujan identities of A003114 and A003106, starting with the constant term 1. The g.f. for the present sequence is G[3](q) = (G[1](q) - G[2](q))/q = 1+q^3+q^4+q^5+q^6+q^7+2*q^8+2*q^9+3*q^10+.... - Joerg Arndt, Oct 08 2012; N. J. A. Sloane, Nov 18 2015

For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[3](x). - N. J. A. Sloane, Nov 22 2015

From Wolfdieter Lang, Oct 31 2016: (Start)

From Hardy (H) p. 94, eq. (6.12.1) and Hardy-Wright (H-W), p. 293, eq. (19.14.3) for H_2(a,x) - H_1(a,x) = a*H_1(a*x,x) one finds from the result for H_1(a,x) (in (H) on top on p. 95), after putting a=x, the o.g.f. of a(n) = A003114(n) - A003106(n), n >= 0, with a(0) = 0 as Sum_{m>=0} x^((m+1)^2) / Product_{j=1..m} (1 - x^j). The m=0 term is 1*x^1. See the formula given by Joerg Arndt, Jan 29 2011.

This formula has a combinatorial interpretation (found similar to the one given in (H) section 6.0, pp. 91-92 or (H-W) pp. 290-291): a(n) is the number of partitions of n with parts differing by at least 2 and part 1 present. See the example for a(15) below. (End)

The Heinz numbers of these integer partitions are given by A324522. - Gus Wiseman, Mar 09 2019

The "least gap" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.

We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.

In the Maple program the subprogram B yields the partition with Heinz number n.

Sum of least gaps of all partitions of m = A022567(m).

From Antti Karttunen, Aug 22 2016: (Start)

Index of the least prime not dividing n. (After a formula given by Heinz.)

Least k such that A002110(k) does not divide n.

One more than the number of trailing zeros in primorial base representation of n, A049345.

(End)

The least gap is also called the mex (minimal excludant) of the partition. - Gus Wiseman, Apr 20 2021

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

Also total number of largest parts in all partitions of n. - Vladeta Jovovic, Feb 16 2004

To see this, consider the properties of a partition related through conjugation, such as the total number of parts and the size of the largest parts. The sums over all of the partitions of n of these two properties are equal. The size of the smallest part and the number of largest parts are two such properties (this is immediate when looking at the Ferrers diagram). - Michael Donatz, Apr 17 2011

Starting with offset 1, = the partition triangle A026794 * [1, 2, 3, ...]. - Gary W. Adamson, Feb 13 2008

For n >= 1, a(n) = T(n+1,1) + T(n+2,2) + T(n+3,3)+ ... (sum along a falling diagonal) of the partition triangle A026794. - Bob Selcoe, Jun 22 2013

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 4 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Jan 2011 and Feb 2012

By removing a single part of size 4, an A026797 partition of n becomes an A008484 partition of n - 4. Hence this sequence is essentially the same as A026797. - Jason Kimberley, Feb 2012

Number of partitions of n+3 such that 3*(number of parts) is a part. - Clark Kimberling, Feb 27 2014

Let c(n) be the number of partitions of n such that both (number of parts) and 2*(number of parts) are parts; then c(n) = a(n-6) for n >= 6 and c(n) = 0 for n < 6. - Clark Kimberling, Mar 01 2014

a(n) is also the number of partitions of n for which three times the number of ones is twice the number of parts (conjectured). - George Beck, Aug 19 2017

Proof: Above definition is equivalent to 2 out of 3 parts being equal to 1. Arrange in triples 1, 1, >= 2, etc. Sum of each triple corresponds to sequence definition. - Martin Fuller, Aug 21 2023

Let b(k) be the number of partitions of k for which twice the number of ones is the number of parts, k = 0, 1, 2, ... . Then a(n+4) = b(n), n = 0, 1, 2, ... (conjectured). - George Beck, Aug 19 2017