A138138 -id:A138138 - cp's OEIS frontend

A138879 Sum of all parts of the last section of the set of partitions of n.

1, 3, 5, 11, 15, 31, 39, 71, 94, 150, 196, 308, 389, 577, 750, 1056, 1353, 1881, 2380, 3230, 4092, 5412, 6821, 8935, 11150, 14386, 17934, 22834, 28281, 35735, 43982, 55066, 67551, 83821, 102365, 126267, 153397, 188001, 227645, 277305, 334383

Offset: 1

Omar E. Pol, Apr 30 2008

nonn

Row sums of the triangles A135010, A138121, A138151 and others related to the section model of partitions (see A135010 and A138121).
From Omar E. Pol, Jan 20 2021: (Start)
Convolution of A000203 and A002865.
Convolution of A340793 and A000041.
Row sums of triangles A339278, A340426, A340583. (End)
a(n) is also the sum of all divisors of all terms of n-th row of A336811. These divisors are also all parts in the last section of the set of partitions of n. - Omar E. Pol, Jul 27 2021
Row sums of A336812. - Omar E. Pol, Aug 03 2021

			a(6)=31 because the parts of the last section of the set of partitions of 6 are (6), (3,3), (4,2), (2,2,2), (1), (1), (1), (1), (1), (1), (1), so the sum is a(6) = 6 + 3 + 3 + 4 + 2 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 31.
From _Omar E. Pol_, Aug 13 2013: (Start)
Illustration of initial terms:
.                                           _ _ _ _ _ _
.                                          |_ _ _ _ _ _|
.                                          |_ _ _|_ _ _|
.                                          |_ _ _ _|_ _|
.                               _ _ _ _ _  |_ _|_ _|_ _|
.                              |_ _ _ _ _|           |_|
.                     _ _ _ _  |_ _ _|_ _|           |_|
.                    |_ _ _ _|         |_|           |_|
.             _ _ _  |_ _|_ _|         |_|           |_|
.       _ _  |_ _ _|       |_|         |_|           |_|
.   _  |_ _|     |_|       |_|         |_|           |_|
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1    3      5        11         15           31
.
(End)
On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the sum of all divisors of these terms is [1 + 2 + 3 + 6] + [1 + 2 + 4] + [1 + 3] + [1 + 2] + [1 + 2] + [1] + [1] = 31, so a(6) = 31. - _Omar E. Pol_, Jul 27 2021

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000041, A000203, A002865, A066186, A133041, A135010, A138121, A138135 - A138138, A138151, A138880, A139100, A237593, A336811, A336812, A338156, A339278, A340035, A340426, A340583, A340793.

A066186 := proc(n) n*combinat[numbpart](n) ; end proc:
A138879 := proc(n) A066186(n)-A066186(n-1) ; end proc:
seq(A138879(n),n=1..80) ; # R. J. Mathar, Jan 27 2011

Table[PartitionsP[n]*n - PartitionsP[n-1]*(n-1), {n, 1, 50}] (* Vaclav Kotesovec, Oct 21 2016 *)

for(n=1, 50, print1(numbpart(n)*n - numbpart(n - 1)*(n - 1),", ")) \\ Indranil Ghosh, Mar 19 2017

from sympy.ntheory import npartitions
print([npartitions(n)*n - npartitions(n - 1)*(n - 1) for n in range(1, 51)]) # Indranil Ghosh, Mar 19 2017

a(n) = A000041(n)*n - A000041(n-1)*(n-1) = A138880(n) + A000041(n-1).
a(n) = A066186(n) - A066186(n-1), for n>=1.
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi/(12*sqrt(2*n)) * (1 - (72 + 13*Pi^2) / (24*Pi*sqrt(6*n)) + (7/12 + 3/(2*Pi^2) + 217*Pi^2/6912)/n - (15*sqrt(3/2)/(16*Pi) + 115*Pi/(288*sqrt(6)) + 4069*Pi^3/(497664*sqrt(6)))/n^(3/2)). - Vaclav Kotesovec, Oct 21 2016, extended Jul 06 2019
G.f.: x*(1 - x)*f'(x), where f(x) = Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017

a(34) corrected by R. J. Mathar, Jan 27 2011

A138880 Sum of all parts of all partitions of n that do not contain 1 as a part.

Original entry on oeis.org

0, 2, 3, 8, 10, 24, 28, 56, 72, 120, 154, 252, 312, 476, 615, 880, 1122, 1584, 1995, 2740, 3465, 4620, 5819, 7680, 9575, 12428, 15498, 19824, 24563, 31170, 38378, 48224, 59202, 73678, 90055, 111384, 135420, 166364, 201630, 246120, 297045, 360822

Offset: 1

Omar E. Pol, Apr 30 2008

nonn

Sum of all parts > 1 of the last section of the set of partitions of n.
Row sums of triangle A182710. Also row sums of other similar triangles as A138136 and A182711.
Partial sums give A194552. - Omar E. Pol, Sep 23 2013

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000041, A002865, A066186, A133041, A138135, A138136, A138137, A138138, A138151, A138879, A139100.

Table[Total[Flatten[Select[IntegerPartitions[n],FreeQ[#,1]&]]],{n,50}] (* Harvey P. Dale, May 24 2015 *)
a[n_] := (PartitionsP[n] - PartitionsP[n-1])*n; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 07 2015 *)

a(n) = A002865(n)*n = (A000041(n) - A000041(n-1))*n = A138879(n) - A000041(n-1).
a(n) ~ Pi^2/6*A000070(n-2). - Peter Bala, Dec 23 2013
G.f.: x*f'(x), where f(x) = Product_{k>=2} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2*n)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (217*Pi^2/6912 + 9/(2*Pi^2) + 13/8)/n). - Vaclav Kotesovec, Jul 06 2019

Better definition from Omar E. Pol, Sep 23 2013

A138136 Triangle read by rows: row n lists the parts > 1 of the last section of the set of partitions of n.

Original entry on oeis.org

0, 2, 3, 4, 2, 2, 5, 3, 2, 6, 4, 2, 3, 3, 2, 2, 2, 7, 5, 2, 4, 3, 3, 2, 2, 8, 6, 2, 5, 3, 4, 4, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 9, 7, 2, 6, 3, 5, 4, 5, 2, 2, 4, 3, 2, 3, 3, 3, 3, 2, 2, 2, 10, 8, 2, 7, 3, 6, 4, 6, 2, 2, 5, 5, 5, 3, 2, 4, 4, 2, 4, 3, 3, 4, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Mar 23 2008

See A138138 and A138151 for more information.

			Triangle begins:
0;
2;
3;
4,2,2;
5,3,2;
6,4,2,3,3,2,2,2;
7,5,2,4,3,3,2,2;
8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,2;
9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2,2;
10,8,2,7,3,6,4,6,2,2,5,5,5,3,2,4,4,2,4,3,3,4,2,2,2,3,3,2,2,2,2,2,2,2;
...

Robert Price, Table of n, a(n) for n = 1..9514, 25 rows.

Cf. A138137, A138138, A138151.

Join[{0}, Table[Cases[IntegerPartitions[n], x_ /; Last[x] != 1], {n, 10}]] // Flatten (* Robert Price, May 22 2020 *)

A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 2, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 1, 6, 4, 2, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 5, 2, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 6, 2, 5, 3, 4, 4, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 7, 2, 6, 3, 5, 4, 5, 2, 2, 4, 3, 2, 3, 3, 3, 3, 2, 2

Offset: 1

Omar E. Pol, Mar 21 2008

The remainder of row n is necessarily A000041(n-1) 1's.
Previous name: A shell model of partitions. Row n lists the parts of the last section of the set of partitions of n.
Row n lists the nonzero terms of the n-th row of A138136 together with A000041(n-1) 1's.
Row n is also the n-th row of A138138 in reverse order.

			Triangle begins:
1
2,1
3,1,1
4,2,2,1,1,1
5,3,2,1,1,1,1,1,
6,4,2,3,3,2,2,2,1,1,1,1,1,1,1
7,5,2,4,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1
8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

Robert Price, Table of n, a(n) for n = 1..4630, 20 rows.
Omar E. Pol, Illustration of the rows 1..6 of the triangle

Mirror of A138138.
Row lengths give A138137.
Row sums give A138879.
Column 1 gives A000027.
Right border gives A000012.
Another version is A138121 which is the mirror of A135010.
Cf. A000041, A080577, A138136, A139094, A139100.

Table[Cases[IntegerPartitions[n], x_ /; Last[x] != 1] ~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 8}] // Flatten (* Robert Price, May 22 2020 *)

New name and comments edited by Peter Munn and Omar E. Pol, Jul 25 2025

A139100 Triangle read by rows: row n lists all partitions of n in the order produced by the shell model of partitions A138151.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 5, 3, 2, 4, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 4, 2, 3, 3, 2, 2, 2, 5, 1, 3, 2, 1, 4, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 2, 4, 3, 3, 2, 2, 6, 1, 4, 2, 1, 3, 3, 1, 2, 2, 2, 1, 5, 1, 1, 3, 2, 1, 1, 4

Offset: 1

Omar E. Pol, Apr 15 2008

See the integrated diagram of partitions in the entry A138138.
See A138151 for more information.
First 43 members = A026792.

			Triangle begins:
{(1)}
{(2), (1, 1)}
{(3), (2, 1), (1, 1, 1)}
{(4), (2, 2), (3, 1), (2, 1, 1), (1, 1, 1, 1)}
{(5), (3, 2), (4, 1), (2, 2, 1), (3, 1, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1)}

Robert Price, Table of n, a(n) for n = 1..7043, 17 rows.

Cf. A000041, A006128, A026792, A036036, A080576, A080577, A135010, A138121, A138135, A138136, A138137, A138138, A138151.

Table[If[n == 1, ConstantArray[{1}, i - n + 1],
   Map[(Join[#, ConstantArray[{1}, i - n]]) &,
    Cases[IntegerPartitions[n], x_ /; Last[x] != 1]]], {i, 7}, {n, i, 1, -1}]  // Flatten(* Robert Price, May 28 2020 *)

A140691 Suppress the zeros in A126441 and resort the natural numbers as illustrated in sequence A136101.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 15, 6, 16, 17, 19, 23, 31, 10, 13, 32, 33, 35, 39, 47, 63, 18, 21, 27, 12, 14, 64, 65, 67, 71, 79, 95, 127, 34, 37, 43, 55, 20, 25, 22, 29, 128, 129, 131, 135, 143, 159, 191, 255, 66, 69, 75, 87, 111, 36, 41, 51, 38, 45, 59, 24, 26, 30, 256, 257, 259

Offset: 1

Alford Arnold, May 23 2008

nonn

Table A138138 (a shell model) is another sequence illustrating row patterns for numeric partitions.

			A136101(11) is 36 and is 2*2*3*3 corresponding to the partition 2+2 which maps to 6 in A136101 so A140691(11) is 6.

Cf. A025487 A138138.

A139094 Largest part of the n-th row in the integrated diagram of the shell model of partitions.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 3, 4, 6, 3, 4, 5, 7, 2, 3, 4, 4, 5, 6, 8, 3, 3, 4, 5, 5, 6, 7, 9, 2, 3, 4, 4, 4, 5, 5, 6, 6, 7, 8, 10, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 9, 11, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 9, 10, 12

Offset: 1

Omar E. Pol, May 26 2008

nonn

Omar E. Pol, A shell model of partitions
Omar E. Pol, Illustration of initial terms

Cf. A138135, A138136, A138137, A138138, A138151, A138879, A138880, A139100.

cp's OEIS Frontend

A138879 Sum of all parts of the last section of the set of partitions of n.

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A138880 Sum of all parts of all partitions of n that do not contain 1 as a part.

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A138136 Triangle read by rows: row n lists the parts > 1 of the last section of the set of partitions of n.

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A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).

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A139100 Triangle read by rows: row n lists all partitions of n in the order produced by the shell model of partitions A138151.

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A140691 Suppress the zeros in A126441 and resort the natural numbers as illustrated in sequence A136101.

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A139094 Largest part of the n-th row in the integrated diagram of the shell model of partitions.

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