|
%I #72 Nov 16 2021 04:16:35
%S 1,1,2,2,4,4,7,7,12,12,19,19,30,30,45,45,67,67,97,97,139,139,195,195,
%T 272,272,373,373,508,508,684,684,915,915,1212,1212,1597,1597,2087,
%U 2087,2714,2714,3506,3506,4508,4508,5763,5763,7338,7338,9296,9296,11732,11732,14742,14742,18460,18460,23025,23025,28629,28629
%N Number of palindromic partitions of n.
%C That is, the number of partitions of n into parts which can be listed in palindromic order.
%C Alternatively, number of partitions of n into parts from the set {1,2,4,6,8,10,12,...}. - _T. D. Noe_, Aug 05 2005
%C Also, partial sums of A035363.
%C Also number of partitions of n with at most one part occurring an odd number of times. - _Reinhard Zumkeller_, Dec 18 2013
%C The first Mathematica program computes terms of A025065; the second computes the k palindromic partitions of user-chosen n. - _Clark Kimberling_, Jan 20 2014
%C a(n) is the number of partitions p of n+1 such that 2*max(p) > n+1. - _Clark Kimberling_, Apr 20 2014.
%C From _Gus Wiseman_, Nov 28 2018: (Start)
%C 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:
%C (41111): {{1,2},{1,3},{1,4},{1,5}}
%C (32111): {{1,2},{1,3},{1,4},{2,5}}
%C (22211): {{1,2},{1,3},{2,4},{3,5}}
%C (311111): {{1,2},{1,3},{1,4,5,6}}
%C (221111): {{1,2},{1,3},{2,4,5,6}}
%C (2111111): {{1,2},{1,3,4,5,6,7}}
%C (11111111): {{1,2,3,4,5,6,7,8}}
%C (End)
%C 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
%C From _Gus Wiseman_, May 21 2021: (Start)
%C 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.
%C 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:
%C (1) (2) (3) (4) (5) (6) (7) (8)
%C (11) (21) (22) (32) (33) (43) (44)
%C (31) (41) (42) (52) (53)
%C (211) (311) (51) (61) (62)
%C (321) (421) (71)
%C (411) (511) (422)
%C (3111) (4111) (431)
%C (521)
%C (611)
%C (4211)
%C (5111)
%C (41111)
%C 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:
%C (1) (2) (21) (22) (221) (222) (2221) (2222)
%C (11) (111) (31) (311) (321) (3211) (3221)
%C (211) (2111) (411) (4111) (3311)
%C (1111) (11111) (2211) (22111) (4211)
%C (3111) (31111) (5111)
%C (21111) (211111) (22211)
%C (111111) (1111111) (32111)
%C (41111)
%C (221111)
%C (311111)
%C (2111111)
%C (11111111)
%C (End)
%H David A. Corneth, <a href="/A025065/b025065.txt">Table of n, a(n) for n = 0..10000</a> (first 101 terms from Reinhard Zumkeller that were corrected by _Georg Fischer_, Jan 20 2019)
%F a(n) = A000070(A004526(n)). - _Reinhard Zumkeller_, Jan 23 2010
%F G.f.: 1/((1-q)*prod(n>=1, 1-q^(2*n))). [_Joerg Arndt_, Mar 11 2014]
%F a(2*k+2) = a(2*k) + A000041(k + 1). - _David A. Corneth_, May 29 2021
%F a(n) ~ exp(Pi*sqrt(n/3)) / (2*Pi*sqrt(n)). - _Vaclav Kotesovec_, Nov 16 2021
%e The partitions for the first few values of n are as follows:
%e n: partitions .......................... number
%e 1: 1 ................................... 1
%e 2: 2 11 ................................ 2
%e 3: 3 111 ............................... 2
%e 4: 4 22 121 1111 ....................... 4
%e 5: 5 131 212 11111 ..................... 4
%e 6: 6 141 33 222 1221 11211 111111 ...... 7
%e 7: 7 151 313 11311 232 21112 1111111 ... 7
%e From _Reinhard Zumkeller_, Jan 23 2010: (Start)
%e Partitions into 1,2,4,6,... for the first values of n:
%e 1: 1 ....................................... 1
%e 2: 2 11 .................................... 2
%e 3: 21 111 .................................. 2
%e 4: 4 22 211 1111 ........................... 4
%e 5: 41 221 2111 11111 ....................... 4
%e 6: 6 42 4211 222 2211 21111 111111.......... 7
%e 7: 61 421 42111 2221 22111 211111 1111111 .. 7. (End)
%t Map[Length[Select[IntegerPartitions[#], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &]] &, Range[40]] (* _Peter J. C. Moses_, Jan 20 2014 *)
%t n = 8; Select[IntegerPartitions[n], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &] (* _Peter J. C. Moses_, Jan 20 2014 *)
%t CoefficientList[Series[1/((1 - x) Product[1 - x^(2 n), {n, 1, 50}]), {x, 0, 60}], x] (* _Clark Kimberling_, Mar 14 2014 *)
%o (Haskell)
%o a025065 = p (1:[2,4..]) where
%o p [] _ = 0
%o p _ 0 = 1
%o p ks'@(k:ks) m | m < k = 0
%o | otherwise = p ks' (m - k) + p ks m
%o -- _Reinhard Zumkeller_, Aug 12 2011
%o (Haskell)
%o import Data.List (group)
%o a025065 = length . filter (<= 1) .
%o map (sum . map ((`mod` 2) . length) . group) . ps 1
%o where ps x 0 = [[]]
%o ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
%o -- _Reinhard Zumkeller_, Dec 18 2013
%o (PARI) N=66; q='q+O('q^N); Vec( 1/((1-q)*eta(q^2)) ) \\ _Joerg Arndt_, Mar 11 2014
%Y Cf. A172033, A004277. - Reinhard Zumkeller, Jan 23 2010
%Y Cf. A004526, A030019, A056503, A147878, A320921, A322136.
%Y The bisections are both A000070.
%Y The ordered version (palindromic compositions) is A016116.
%Y The complement is counted by A233771 and A210249.
%Y The case of palindromic prime signature is A242414.
%Y Palindromic partitions are ranked by A265640, with complement A229153.
%Y The case of palindromic plane trees is A319436.
%Y The multiplicative version (palindromic factorizations) is A344417.
%Y A000569 counts graphical partitions.
%Y A027187 counts partitions of even length, ranked by A028260.
%Y A035363 counts partitions into even parts, ranked by A066207.
%Y A058696 counts partitions of even numbers, ranked by A300061.
%Y A110618 counts partitions with length <= half sum, ranked by A344291.
%Y Cf. A000041, A067538, A143773, A209816, A338914, A338915, A340387, A344296, A344414, A344415, A344416.
%K nonn
%O 0,3
%A _Clark Kimberling_
%E Edited by _N. J. A. Sloane_, Dec 29 2007
%E Prepended a(0)=1, added more terms, _Joerg Arndt_, Mar 11 2014
| 