A025065 Number of palindromic partitions of n.

1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296, 11732, 11732, 14742, 14742, 18460, 18460, 23025, 23025, 28629, 28629

Offset

0,3

Comments

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)

Links

David A. Corneth, Table of n, a(n) for n = 0..10000 (first 101 terms from Reinhard Zumkeller that were corrected by Georg Fischer, Jan 20 2019)

Formula

a(n) = A000070(A004526(n)). - Reinhard Zumkeller, Jan 23 2010

G.f.: 1/((1-q)*prod(n>=1, 1-q^(2*n))). [Joerg Arndt, Mar 11 2014]

a(2*k+2) = a(2*k) + A000041(k + 1). - David A. Corneth, May 29 2021

a(n) ~ exp(Pi*sqrt(n/3)) / (2*Pi*sqrt(n)). - Vaclav Kotesovec, Nov 16 2021

Example

The partitions for the first few values of n are as follows:
n: partitions .......................... number
1: 1 ................................... 1
2: 2 11 ................................ 2
3: 3 111 ............................... 2
4: 4 22 121 1111 ....................... 4
5: 5 131 212 11111 ..................... 4
6: 6 141 33 222 1221 11211 111111 ...... 7
7: 7 151 313 11311 232 21112 1111111 ... 7
From Reinhard Zumkeller, Jan 23 2010: (Start)
Partitions into 1,2,4,6,... for the first values of n:
1: 1 ....................................... 1
2: 2 11 .................................... 2
3: 21 111 .................................. 2
4: 4 22 211 1111 ........................... 4
5: 41 221 2111 11111 ....................... 4
6: 6 42 4211 222 2211 21111 111111.......... 7
7: 61 421 42111 2221 22111 211111 1111111 .. 7. (End)

Mathematica

Map[Length[Select[IntegerPartitions[#], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &]] &, Range[40]] (* Peter J. C. Moses, Jan 20 2014 *)
n = 8; Select[IntegerPartitions[n], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &] (* Peter J. C. Moses, Jan 20 2014 *)
CoefficientList[Series[1/((1 - x) Product[1 - x^(2 n), {n, 1, 50}]), {x, 0, 60}], x] (* Clark Kimberling, Mar 14 2014 *)

Prog

(Haskell)
a025065 = p (1:[2, 4..]) where
   p [] _ = 0
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011
(Haskell)
import Data.List (group)
a025065 = length . filter (<= 1) .
                   map (sum . map ((`mod` 2) . length) . group) . ps 1
   where ps x 0 = [[]]
         ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 18 2013
(PARI) N=66; q='q+O('q^N); Vec( 1/((1-q)*eta(q^2)) ) \\ Joerg Arndt, Mar 11 2014

Crossrefs

Cf. A172033, A004277. - Reinhard Zumkeller, Jan 23 2010

Cf. A004526, A030019, A056503, A147878, A320921, A322136.

The bisections are both A000070.

The ordered version (palindromic compositions) is A016116.

The complement is counted by A233771 and A210249.

The case of palindromic prime signature is A242414.

Palindromic partitions are ranked by A265640, with complement A229153.

The case of palindromic plane trees is A319436.

The multiplicative version (palindromic factorizations) is A344417.

A000569 counts graphical partitions.

A027187 counts partitions of even length, ranked by A028260.

A035363 counts partitions into even parts, ranked by A066207.

A058696 counts partitions of even numbers, ranked by A300061.

A110618 counts partitions with length <= half sum, ranked by A344291.

Cf. A000041, A067538, A143773, A209816, A338914, A338915, A340387, A344296, A344414, A344415, A344416.

Sequence in context: A371514 A363214 A062896 * A306664 A365826 A131524

Adjacent sequences: A025062 A025063 A025064 * A025066 A025067 A025068

Keyword

nonn

Author

Clark Kimberling

Extensions

Edited by N. J. A. Sloane, Dec 29 2007

Prepended a(0)=1, added more terms, Joerg Arndt, Mar 11 2014

Status

approved