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A025065 Number of palindromic partitions of n. 51
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 (list; graph; refs; listen; history; text; internal format)
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 such that 2*max(p) > n, for 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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..100 (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]

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. A000070, A000569, A004526, A030019, A056503, A110618, A147878, A209816, A242414, A319436, A320921, A322136.

Sequence in context: A304178 A266776 A062896 * A306664 A131524 A089075

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

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Last modified March 30 09:13 EDT 2020. Contains 333125 sequences. (Running on oeis4.)