cp's OEIS Frontend

Essentially the same as A001700, which has more information.

Note that the unique rooted tree with no edges has no leaves, so a(0)=1 is by convention. - Michael Somos, Jul 30 2011

Number of ordered partitions of n into n parts, allowing zeros (cf. A097070) is binomial(2*n-1,n) = a(n) = essentially A001700. - Vladeta Jovovic, Sep 15 2004

Hankel transform is A000027; example: Det([1,1,3,10;1,3,10,35;3,10,35,126; 10,35,126,462]) = 4. - Philippe Deléham, Apr 13 2007

a(n) is the number of functions f:[n]->[n] such that for all x,y in [n] if xA045992(n). - Geoffrey Critzer, Apr 02 2009

Hankel transform of the aeration of this sequence is A000027 doubled: 1,1,2,2,3,3,... - Paul Barry, Sep 26 2009

The Fi1 and Fi2 triangle sums of A039599 are given by the terms of this sequence. For the definitions of these triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011

Alternating row sums of Riordan triangle A094527. See the Philippe Deléham formula. - Wolfdieter Lang, Nov 22 2012

(-2)*a(n) is the Z-sequence for the Riordan triangle A110162. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. - Wolfdieter Lang, Nov 22 2012

From Gus Wiseman, Jun 27 2021: (Start)

Also the number of integer compositions of 2n with alternating (or reverse-alternating) sum 0 (ranked by A344619). This is equivalent to Ran Pan's comment at A001700. For example, the a(0) = 1 through a(3) = 10 compositions are:

() (11) (22) (33)

(121) (132)

(1111) (231)

(1122)

(1221)

(2112)

(2211)

(11121)

(12111)

(111111)

For n > 0, a(n) is also the number of integer compositions of 2n with alternating sum 2.

(End)

Number of terms in the expansion of (x_1+x_2+...+x_n)^n. - César Eliud Lozada, Jan 08 2022

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

For n > 0, also the number of partitions of n whose greatest part is even. [Edited by Gus Wiseman, Jan 05 2021]

Number of partitions of n+1 into an odd number of parts, the least being 1.

Also the number of partitions of n such that the number of even parts has the same parity as the number of odd parts; see Comments at A027193. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021

Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1). When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k). Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n. Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts. That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms. The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

That is, the number of partitions of n into parts which can be listed in palindromic order.

Alternatively, number of partitions of n into parts from the set {1,2,4,6,8,10,12,...}. - T. D. Noe, Aug 05 2005

Also, partial sums of A035363.

Also number of partitions of n with at most one part occurring an odd number of times. - Reinhard Zumkeller, Dec 18 2013

The first Mathematica program computes terms of A025065; the second computes the k palindromic partitions of user-chosen n. - Clark Kimberling, Jan 20 2014

a(n) is the number of partitions p of n+1 such that 2*max(p) > n+1. - Clark Kimberling, Apr 20 2014.

From Gus Wiseman, Nov 28 2018: (Start)

Also the number of integer partitions of n + 2 that are the vertex-degrees of some hypertree. For example, the a(6) = 7 partitions of 8 that are the vertex-degrees of some hypertree, together with a realizing hypertree are:

(41111): {{1,2},{1,3},{1,4},{1,5}}

(32111): {{1,2},{1,3},{1,4},{2,5}}

(22211): {{1,2},{1,3},{2,4},{3,5}}

(311111): {{1,2},{1,3},{1,4,5,6}}

(221111): {{1,2},{1,3},{2,4,5,6}}

(2111111): {{1,2},{1,3,4,5,6,7}}

(11111111): {{1,2,3,4,5,6,7,8}}

(End)

Conjecture: a(n) is the length of maximal initial segment of A308355(n-1) that is identical to row n of A128628, for n >= 2. - Clark Kimberling, May 24 2019

From Gus Wiseman, May 21 2021: (Start)

The Heinz numbers of palindromic partitions are given by A265640. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

Also the number of integer partitions of n with a part greater than or equal to n/2. This is equivalent to Clark Kimberling's final comment above. The Heinz numbers of these partitions are given by A344414. For example, the a(1) = 1 through a(8) = 12 partitions are:

(1) (2) (3) (4) (5) (6) (7) (8)

(11) (21) (22) (32) (33) (43) (44)

(31) (41) (42) (52) (53)

(211) (311) (51) (61) (62)

(321) (421) (71)

(411) (511) (422)

(3111) (4111) (431)

(521)

(611)

(4211)

(5111)

(41111)

Also the number of integer partitions of n with at least n/2 parts. The Heinz numbers of these partitions are given by A344296. For example, the a(1) = 1 through a(8) = 12 partitions are:

(1) (2) (21) (22) (221) (222) (2221) (2222)

(11) (111) (31) (311) (321) (3211) (3221)

(211) (2111) (411) (4111) (3311)

(1111) (11111) (2211) (22111) (4211)

(3111) (31111) (5111)

(21111) (211111) (22211)

(111111) (1111111) (32111)

(41111)

(221111)

(311111)

(2111111)

(11111111)

(End)

Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009

Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009

Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013

Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016

From Gus Wiseman, May 22 2021: (Start)

The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290.

For n > 1, also the number of integer partitions of n-1 whose only odd part is the smallest. The Heinz numbers of these partitions are given by A341446. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns shown as dots, A..D = 10..13) are:

1 . 3 . 5 . 7 . 9 . B . D

21 41 43 63 65 85

221 61 81 83 A3

421 441 A1 C1

2221 621 443 643

4221 641 661

22221 821 841

4421 A21

6221 4441

42221 6421

222221 8221

44221

62221

422221

2222221

Also the number of integer partitions of n whose greatest part is the sum of all the other parts. The Heinz numbers of these partitions are given by A344415. For example, the a(2) = 1 through a(12) = 11 partitions (empty columns not shown) are:

(11) (22) (33) (44) (55) (66)

(211) (321) (422) (532) (633)

(3111) (431) (541) (642)

(4211) (5221) (651)

(41111) (5311) (6222)

(52111) (6321)

(511111) (6411)

(62211)

(63111)

(621111)

(6111111)

Also the number of integer partitions of n of length n/2. The Heinz numbers of these partitions are given by A340387. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns not shown) are:

(2) (22) (222) (2222) (22222) (222222) (2222222)

(31) (321) (3221) (32221) (322221) (3222221)

(411) (3311) (33211) (332211) (3322211)

(4211) (42211) (333111) (3332111)

(5111) (43111) (422211) (4222211)

(52111) (432111) (4322111)

(61111) (441111) (4331111)

(522111) (4421111)

(531111) (5222111)

(621111) (5321111)

(711111) (5411111)

(6221111)

(6311111)

(7211111)

(8111111)

(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

A bisection of A000041, the other one is A058696.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). - Michael Somos, Feb 16 2014

a(n) is the number of partitions of 3n-1 having n as a part, for n >=1. Also, a(n+1) is the number of partitions of 3n having n as a part, for n >= 1. - Clark Kimberling, Mar 02 2014

Interleaving of A114121 and A032443. Row sums of A116405. Binomial transform is A116409.

Appears to be the number of n-digit binary numbers not having more zeros than ones; equivalently, the number of unrestricted Dyck paths of length n not going below the axis. - Ralf Stephan, Mar 25 2008

From Gus Wiseman, Jun 20 2021: (Start)

Also the number compositions of n with alternating sum >= 0, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. The a(0) = 1 through a(5) = 11 compositions are:

() (1) (2) (3) (4) (5)

(11) (21) (22) (32)

(111) (31) (41)

(112) (113)

(121) (122)

(211) (212)

(1111) (221)

(311)

(1121)

(2111)

(11111)

(End)

From J. Stauduhar, Jan 14 2022: (Start)

Also, for n >= 2, first differences of partial row sums of Pascal's triangle. The first ceiling(n/2)+1 elements of rows n=0 to n=4 in Pascal's triangle are:

1

1 1

1 2

1 3 3

1 4 6

...

The cumulative sums of these partial rows form the sequence 1,3,6,13,24,..., and its first differences are a(2),a(3),a(4),... in this sequence.

(End)

Ramanujan theta functions: phi(q) (A000122), chi(q) (A000700).

The partitions of n are partitioned into four types:

EO, even # of odd parts and odd # of even parts, A236559;

OE, odd # of odd parts and even # of even parts, A160786;

EE, even # of odd parts and even # of even parts, A236913;

OO, odd # of odd parts and odd # of even parts, A236914.

A236559 and A160786 are the bisections of A027193;

A236913 and A236914 are the bisections of A027187.