A053445 -id:A053445 - cp's OEIS frontend

A129306 Resort sequence A096443 by source partition as described by A053445 and A126442.

1, 2, 2, 3, 4, 5, 5, 7, 11, 15, 9, 7, 12, 21, 36, 52, 16, 26, 11, 19, 38, 74, 135, 203, 29, 52, 92, 31, 66, 15, 30, 64, 141, 296, 566, 877, 47, 98, 198, 371, 57, 109, 137, 249, 22, 45, 105, 250, 592, 1315, 2610, 4140, 77, 171, 392, 850, 1663, 97, 212, 444, 269, 560, 1075

Offset: 1

Alford Arnold, May 07 2007

nonn

The first array is described in A126442 and is the hook case. Sequence A129305 encodes the multisets counted by A096443 and A129306.

			a(11) = 9 because 2+2= 4 starting a new array. The arrays begin as follows:
1.....2.....3.....5......7......11.....15.....22
......2.....4.....7......12.....19.....30.....45
............5.....11.....21.....38.....64.....105
..................15.....36.....74.....141....250
.........................52.....135....296....592
................................203....566....1315
.......................................877....2610
..............................................4140
..................9......16.....29.....47.....77
.........................26.....52.....98.....171
................................92.....198....392
.......................................371....850
..............................................1663
................................31.....57.....97
.......................................109....212
..............................................444
................................66.....137....269
.......................................249....560
..............................................1075
..............................................109
..............................................300
..............................................712
which sums to
1.....4....12....47....170....750....3255.....16010

Cf. A000041, A035310, A035341, A053445, A054225, A082775, A093802, A096443, A126442, A129305.

A160644 First of two sequences bisecting the second differences of the partition numbers (see A053445).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 14, 22, 32, 45, 67, 95, 134, 192, 269, 373, 521, 718, 983, 1346, 1827, 2465, 3323, 4449, 5929, 7882, 10426, 13735, 18047, 23613, 30788, 40034, 51877, 67013, 86341, 110905, 142063, 181529, 231340, 294077, 372977, 471908, 595725, 750432

Offset: 1

Alford Arnold, May 21 2009

A160644 also counts selected unrestricted partition having an EVEN total and with minimum part two. For example it counts these three partitions of eight: 4+4, 3+3+2, and 2+2+2+2.

			A053445 begins
1 0 1 0 2 0 3 1 4 2 7 3 10 7 14 11 22 17 32 28 45 ...
therefore a(n) begins
1 1 2 3 4 7 10 14 22 32 45 67 95 134 192 ...

Nathaniel Johnston, Table of n, a(n) for n = 1..2500

Cf. A000041, A002685, A053445, A160643, A161921 (the other bisection).

Take[Differences[PartitionsP[Range[0,100]],2],{1,-1,2}] (* Harvey P. Dale, Aug 05 2019 *)

A160643 Bisect A053445 then calculate the first differences of the resulting sequence.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 4, 6, 11, 15, 20, 33, 43, 60, 88, 119, 160, 226, 300, 404, 549, 727, 961, 1283, 1680, 2201, 2887, 3750, 4857, 6301, 8105, 10410, 13357, 17050, 21714, 27625, 34992, 44240, 55840, 70261, 88220, 110600, 138274, 172558, 214984, 267234

Offset: 1

Alford Arnold, May 25 2009, Jun 20 2009

a(n) counts the following subset of the partitions (cf. A000041): the number being partitioned is odd, the minimum part is two
and the three largest parts match.
First differences of A161921.

			A161921 begins: 0, 0, 1, 2, 3, 7, 11, 17, 28, 43, 63, 96, 139, 199, 287, 406, 566, ...
Therefore a(n) begins 0, 0, 0, 1, 1, 1, 4, 4, 6, ..., counting 333; 3332; 33322; 555, 4443, 333222, 33333; etc.

Nathaniel Johnston, Table of n, a(n) for n = 1..2500

Cf. A053445, A160644.

Join[{0},Differences[Take[Differences[Table[PartitionsP[n],{n,0,100}],2],{2,-1,2}]]] (* Harvey P. Dale, Sep 02 2013 *)

Extended and edited by Nathaniel Johnston, Apr 30 2011

A161921 The bisection A053445(2n+1).

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 7, 11, 17, 28, 43, 63, 96, 139, 199, 287, 406, 566, 792, 1092, 1496, 2045, 2772, 3733, 5016, 6696, 8897, 11784, 15534, 20391, 26692, 34797, 45207, 58564, 75614, 97328, 124953, 159945, 204185, 260025, 330286, 418506, 529106, 667380, 839938

Offset: 0

Alford Arnold, Jul 05 2009

Nathaniel Johnston, Table of n, a(n) for n = 0..2500

Cf. A160644 (the other bisection), A160643 (first differences of a(n)).

Take[Differences[Table[PartitionsP[n],{n,0,100}],2],{2,-1,2}] (* Harvey P. Dale, Sep 02 2013 *)

A162932 a(n) = A053445(n-2) - A053445(n-4).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 1, 3, 4, 4, 4, 8, 6, 10, 11, 13, 15, 22, 20, 28, 33, 39, 43, 58, 60, 77, 88, 104, 119, 148, 160, 197, 226, 265, 300, 363, 404, 481, 549, 638, 727, 858, 961, 1126, 1283, 1480, 1680, 1953, 2201, 2544, 2887, 3309, 3750, 4312, 4857, 5566, 6301, 7175

Offset: 6

Alford Arnold, Jul 17 2009

nonn

a(n) counts partitions of n such that all parts are >=2 and the largest part occurs at least three times, see example.

			For n = 19 the a(19) = 6 partitions are 5554, 44443, 55522, 444322, 3333322 and 33322222.

Andrew van den Hoeven, Table of n, a(n) for n = 6..10000

Cf. A162931, A079946.

a:=func; [a(n): n in [6..100]]; // Vincenzo Librandi, Dec 09 2014

Table[PartitionsP[n] - 2 PartitionsP[n - 1] + 2 PartitionsP[n - 3] - PartitionsP[n - 4], {n, 6, 70}] (* Vincenzo Librandi, Dec 09 2014 *)

From Mircea Merca, Jun 11 2012: (Start)
a(n) = p(n) - 2*p(n-1) + 2*p(n-3) - p(n-4) for n >= 6, where p(n) = A000041(n).
G.f.: -1 + x - x^3 + (1 - x)*Product_{k > 2} 1/(1 - x^k). (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (36*sqrt(2)*n^(5/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{n >= 1} q^(3*n+3)/Product_{k = 1..n} 1 - q^(k+1). - Peter Bala, Dec 01 2024

Keyword:tabf removed, indexing corrected, sequence extended by R. J. Mathar, Sep 17 2009

A136101 Resort sequence A025487 by source partition as described by A053445 and A126442.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 30, 16, 24, 60, 210, 36, 32, 48, 120, 420, 2310, 72, 180, 64, 96, 240, 840, 4620, 30030, 144, 360, 1260, 216, 900, 128, 192, 480, 1680, 9240, 60060, 510510, 288, 720, 2520, 13860, 432, 1080, 1800, 6300, 256, 384, 960, 3360, 18480, 120120

Offset: 0

Alford Arnold, Jan 01 2008

			Let a(0) = 1 then let the irregular table begin:
2....4....8....16....32....64....
....6...12....24....48....96....
........30....60...120...240....
.............210...420...840....
..................2310..4620....
.......................30030....
..............36....72...144....
...................180...360....
........................1260....
.........................216....
.........................900....

Cf. A025487 A053445 A126442 A129306.

A140760 Irregular table of natural numbers (read by columns) which can be mapped to the source partitions described in A053445 and A126441.

Original entry on oeis.org

2, 6, 12, 14, 24, 26, 30, 28, 48, 50, 54, 62, 52, 58, 96, 56, 60, 98, 102, 110, 126, 100, 106, 118, 192, 104, 108, 194, 114, 122, 198, 206, 222, 254, 112, 124, 116, 196, 202, 214, 238, 384, 120, 200, 204, 386, 210, 218, 390, 230, 246, 398, 414, 446, 510

Offset: 1

Alford Arnold, May 28 2008

Sequences A000041, A002865 and A053445 count numeric partitions. A125106 maps numeric partitions to the natural numbers and has A000120 elements per row. A126441 has A000041 elements per column and is a tabular arrangement of the natural numbers. A140691 is a rearrangement of table A126441. A140692 extracts the cyclic cases and has A002865 elements per column.
The values for A140760 can also be generated by beginning with A140759 and repeatedly multiplying by two as follows:
2
....6.....12......24.....48.....96
..........14..........28........56
..................26........52....
..................30............60
.........................50.......
.........................54.......
.........................62.......
Note that the number of entries in each column is given by A053445.

			The values of A140760 can be obtained by selecting the first even number on each applicable row of Table A126441.
Table A126441 begins
1.2.4..8.16.32.64.128.256.512.1024
..3.5..9.17.33.65.129.257.513.1025
.......6.10.18.34..66.130.258..514
....7.11.19.35.67.131.259.515.1027
............12.20..36..68.132..260
.........13.21.37..69.133.261..517
............14.22..38..70.134..262
......15.23.39.71.135.263.519.1031
...................24..40..72..136
...............25..41..73.137..265
...................26..42..74..138
............27.43..75.139.267..523
.......................28..44...76
...............29..45..77.141..269
...................30..46..78..142
.........31.47.79.143.271.527.1039
...........................48...80
.......................49..81..145
...........................50...82
...................51..83.147..275

Cf. A000041, A000120, A002865, A053445, A125106, A126441, A140691, A140692, A140759.

A140759 An irregular table of natural numbers used to generate the values in sequence A140760; which, in turn, can be mapped to the source partitions counted by A053445.

Original entry on oeis.org

2, 6, 14, 26, 30, 50, 54, 62, 58, 98, 102, 110, 126, 106, 118, 194, 114, 122, 198, 206, 222, 254, 202, 214, 238, 386, 210, 218, 390, 230, 246, 398, 414, 446, 510

Offset: 1

Alford Arnold, May 27 2008

nonn

Cf. A053445 A126441 A140760.

A161162 An irregular array distributing the values of A053445 having powers of two as column sums. (cf. A011782).

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 1, 4, 2, 2, 5, 3, 4, 6, 3, 4, 1, 6, 7, 6, 5, 6, 8, 8, 2, 9, 6, 2, 11, 10, 9, 9, 12, 7, 7, 16, 12, 10, 4, 16, 15, 8, 3, 18, 21, 14, 11, 1, 12, 23, 18, 9, 12, 29, 26, 16, 12, 8, 27, 30, 21, 10, 6, 26, 40, 31, 18, 13, 2, 23, 42, 37, 24, 11, 2, 21, 48, 51, 36, 20, 14, 15, 44

Offset: 1

Alford Arnold, Jun 08 2009

Diagonals match the reverse of the rows in sequence A083480.
Contribution from Alford Arnold, Jun 09 2009: (Start)
Sequence A161162 is also closely related to sequence A161161 as illustrated.
consider the two vectors in A161161 summing to 32 and 16 respectively:
1 1 2 3 5 7 5 4 3 1
1 1 2 3 5 2 2
they differ by
0 0 0 0 0 5 3 4 3 1 which is the vector in A161162 summing to 16.
(End)

			Column six is 5 3 4 3 1 and sums to 16.
The Tabf begins 1
.0
.1
..0
..2
...0
...3
...1
....4
....2
....2.5

A161161 [From Alford Arnold, Jun 09 2009]

A126440 Triangular array read by rows: related to A053445 and A060351 with row sums A000142 (which counts permutations of n objects).

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 3, 13, 8, 0, 4, 42, 58, 16, 0, 5, 118, 344, 221, 32

Offset: 1

Alford Arnold, Jan 13 2007

A060351 relates permutations to numeric partitions. Diagonal 1 2 4 8 16 32 ... A000079 sums triangle A007318 and is related to hook partitions. Diagonal 2 13 58 221 ... sums 2 10 34 98 ... related to partition 2+2 3 20 85 ... related to 3+3 4 33 ... related to 4+4 5 ... related to 5+5 etc.

			The array begins
1
0 2
0 2 4
0 3 13 8
0 4 42 58 16
0 5 118 344 221 32

Cf. A000041 A000142 A053445 A059797 A060351 A091307.

cp's OEIS Frontend

A129306 Resort sequence A096443 by source partition as described by A053445 and A126442.

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A160644 First of two sequences bisecting the second differences of the partition numbers (see A053445).

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A160643 Bisect A053445 then calculate the first differences of the resulting sequence.

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A161921 The bisection A053445(2n+1).

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A162932 a(n) = A053445(n-2) - A053445(n-4).

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A136101 Resort sequence A025487 by source partition as described by A053445 and A126442.

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A140760 Irregular table of natural numbers (read by columns) which can be mapped to the source partitions described in A053445 and A126441.

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A140759 An irregular table of natural numbers used to generate the values in sequence A140760; which, in turn, can be mapped to the source partitions counted by A053445.

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A161162 An irregular array distributing the values of A053445 having powers of two as column sums. (cf. A011782).

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A126440 Triangular array read by rows: related to A053445 and A060351 with row sums A000142 (which counts permutations of n objects).

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