A138151 -id:A138151 - cp's OEIS frontend

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

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 *)

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

Original entry on oeis.org

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 *)

A182711 Triangle read by rows in which row n lists the parts > 1 of the last section of the set of partitions of n in an order similar to A138136 but in this case the partitions with the least number of parts are listed first.

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, 5, 5, 6, 2, 2, 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, Nov 29 2010

In this sequence a(68)=5 but in A138136 a(68)=6. See the 8th term in row 10 of triangle.

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

Cf. A135010, A138121, A138136, A138151, A182710.

Row sums give A138880.

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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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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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A182711 Triangle read by rows in which row n lists the parts > 1 of the last section of the set of partitions of n in an order similar to A138136 but in this case the partitions with the least number of parts are listed first.

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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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