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(n) = degree of Kac determinant at level n as polynomial in the conformal weight (called h). (Cf. C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Vol. 2, p. 533, eq.(98); reference p. 643, Cambridge University Press, (1989).) - Wolfdieter Lang

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 from any part z > 1 one can take an element of amount 1 in one way only. That means z is composed of z unlabeled parts of amount 1, i.e. z = 1 + 1 + ... + 1. E.g., for n=3 to n=2 we have a(3) = 6 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. For the case of z composed by labeled elements, z = 1_1 + 1_2 + ... + 1_z, see A066186. - Thomas Wieder, May 20 2004

Number of times a derivative of any order (not 0 of course) appears when expanding the n-th derivative of 1/f(x). For instance (1/f(x))'' = (2 f'(x)^2-f(x) f''(x)) / f(x)^3 which makes a(2) = 3 (by counting k times the k-th power of a derivative). - Thomas Baruchel, Nov 07 2005

Starting with offset 1, = the partition triangle A008284 * [1, 2, 3, ...]. - Gary W. Adamson, Feb 13 2008

Starting with offset 1 equals A000041: (1, 1, 2, 3, 5, 7, 11, ...) convolved with A000005: (1, 2, 2, 3, 2, 4, ...). - Gary W. Adamson, Jun 16 2009

Apart from initial 0 row sums of triangle A066633, also the Möbius transform is A085410. - Gary W. Adamson, Mar 21 2011

More generally, the total number of parts >= k in all partitions of n equals the sum of k-th largest parts of all partitions of n. In this case k = 1. Apart from initial 0 the first column of A181187. - Omar E. Pol, Feb 14 2012

Row sums of triangle A221530. - Omar E. Pol, Jan 21 2013

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

a(n) is also the total number of 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 as follows:

Convolution of A000005 and A000041.

Convolution of A006218 and A002865.

Convolution of A341062 and A000070.

Row sums of triangles A221531, A245095, A339258, A340525, A340529. (End)

Number of ways to choose a part index of an integer partition of n, i.e., partitions of n with a selected position. Selecting a part value instead of index gives A000070. - Gus Wiseman, Apr 19 2021

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)

It appears that row n lists the first differences of the row n of triangle A181187 together with 1 (as the final term of the row n). - Omar E. Pol, Feb 26 2012

It appears that reversed rows converge to A000041. - Omar E. Pol, Mar 11 2012

Proof: For a partition of n with k>floor(n/2+1), k can only occur as the largest part; the other parts sum to n-k, so that T(n,n-k)=A000041(k). - George Beck, Jun 30 2019

T(n,k) is also the total number k's that are 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. - Omar E. Pol, Feb 05 2021