A138135 -id:A138135 - cp's OEIS frontend

A138137 First differences of A006128.

1, 2, 3, 6, 8, 15, 19, 32, 42, 64, 83, 124, 157, 224, 288, 395, 502, 679, 854, 1132, 1422, 1847, 2307, 2968, 3677, 4671, 5772, 7251, 8908, 11110, 13572, 16792, 20439, 25096, 30414, 37138, 44798, 54389, 65386, 78959, 94558, 113687, 135646, 162375, 193133

Offset: 1

Omar E. Pol, Mar 18 2008

Number of parts in the last section of the set of partitions of n (see A135010, A138121).
Sum of largest parts in all partitions in the head of the last section of the set of partitions of n. - Omar E. Pol, Nov 07 2011
From Omar E. Pol, Feb 16 2021: (Start)
Convolution of A341062 and A000041.
Convolution of A000005 and A002865.
a(n) is also the total number of parts in the n-th section of the set of partitions of any positive integer >= n.
a(n) is also the total number of divisors of all terms in the n-th row of triangle A336811. These divisors are also all parts in the last section of the set of partitions of n. (End)

			From _Omar E. Pol_, Feb 19 2012: (Start)
Illustration of initial terms (n = 1..6) as sums of the first columns from the last sections of the first six natural numbers (or from the first six sections of 6):
.                                           6
.                                           3+3
.                                           4+2
.                                           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
. --- ----- ------- --------- ----------- --------------
.  1,  2,    3,      6,        8,          15,
...
Also, we can see that the sequence gives the number of parts in each section. For the number of odd/even parts (and more) see A207031, A207032 and also A206563. (End)
From _Omar E. Pol_, Aug 16 2013: (Start)
The geometric model looks like this:
.                                           _ _ _ _ _ _
.                                          |_ _ _ _ _ _|
.                                          |_ _ _|_ _ _|
.                                          |_ _ _ _|_ _|
.                               _ _ _ _ _  |_ _|_ _|_ _|
.                              |_ _ _ _ _|           |_|
.                     _ _ _ _  |_ _ _|_ _|           |_|
.                    |_ _ _ _|         |_|           |_|
.             _ _ _  |_ _|_ _|         |_|           |_|
.       _ _  |_ _ _|       |_|         |_|           |_|
.   _  |_ _|     |_|       |_|         |_|           |_|
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1    2      3        6          8           15
.
(End)
On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the divisors of these terms are [1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]. There are 15 divisors so a(6) = 15. - _Omar E. Pol_, Jul 27 2021

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Column 1 of A207031.

Cf. A000005, A000041, A002865, A006128, A135010, A138121, A182703, A336811, A336812, A338156, A340035, A341062.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 19 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]]+g[[1]], f[[2]]+g[[2]]+g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
Table[PartitionsP[n - 1] + Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 45}] (* Robert Price, May 01 2020 *)

a(n) = A006128(n) - A006128(n-1).
a(n) = A000041(n-1) + A138135(n). - Omar E. Pol, Nov 07 2011
a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(6*n/Pi^2)) / (8*sqrt(3)*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 21 2016
G.f.: Sum_{i>=1} i*x^i * Product_{j=2..i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 04 2017

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

A207031 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the last section of the set of partitions of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 6, 3, 1, 1, 8, 3, 2, 1, 1, 15, 8, 4, 2, 1, 1, 19, 8, 5, 3, 2, 1, 1, 32, 17, 9, 6, 3, 2, 1, 1, 42, 20, 13, 7, 5, 3, 2, 1, 1, 64, 34, 19, 13, 8, 5, 3, 2, 1, 1, 83, 41, 26, 16, 11, 7, 5, 3, 2, 1, 1, 124, 68, 41, 27, 17, 12, 7, 5, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Feb 14 2012

Also T(n,k) is the number of parts >= k in the last section of the set of partitions of n. Therefore T(n,1) = A138137(n), the total number of parts in the last section of the set of partitions of n. For calculation of the number of odd/even parts, etc, follow the same rules from A206563.
More generally, let m and n be two positive integers such that m <= n. It appears that any set formed by m connected sections, or m disconnected sections, or a mixture of both, has the same properties described in the entry A206563.
It appears that reversed rows converge to A000041.
It appears that the first differences of row n together with 1 give the row n of triangle A182703 (see example). - Omar E. Pol, Feb 26 2012

			Illustration of initial terms. First six rows of triangle as sums of columns from the last sections of the first six natural numbers (or as sums of columns from the six sections of 6):
.                                         6
.                                         3 3
.                                         4 2
.                                         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
. --- --- ------- --------- ----------- --------------
A: 1, 2,1, 3,1,1,  6,3,1,1,  8,3,2,1,1,  15,8,4,2,1,1
.  |  |/|  |/|/|   |/|/|/|   |/|/|/|/|    |/|/|/|/|/|
B: 1, 1,1, 2,0,1,  3,2,0,1,  5,1,1,0,1,   7,4,2,1,0,1
.
A := initial terms of this triangle.
B := initial terms of triangle A182703.
.
Triangle begins:
1;
2,    1;
3,    1,  1;
6,    3,  1,  1;
8,    3,  2,  1,  1;
15,   8,  4,  2,  1,  1;
19,   8,  5,  3,  2,  1,  1;
32,  17,  9,  6,  3,  2,  1,  1;
42,  20, 13,  7,  5,  3,  2,  1,  1;
64,  34, 19, 13,  8,  5,  3,  2,  1,  1;
83,  41, 26, 16, 11,  7,  5,  3,  2,  1,  1;
124, 68, 41, 27, 17, 12,  7,  5,  3,  2,  1,  1;

Columns (1-2): A138137, A138135.
Row sums give A138879.
Cf. A000041, A002865, A006128, A135010, A138121, A181187, A182703, A206562, A206563, A207032, A207379, A208476, A210955, A210956.

From Omar E. Pol, Dec 07 2019: (Start)
From the formula in A138135 (year 2008) we have that:
A000041(n-1) = A138137(n) - A138135(n) = T(n,1) - T(n,2);
Hence A000041(n) = T(n+1,1) - T(n+1,2), n >= 0;
Also  A000041(n) = A002865(n) + T(n,1) - T(n,2). (End)

More terms from Alois P. Heinz, Feb 17 2012

A182699 Number of emergent parts in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 4, 4, 10, 12, 22, 27, 47, 56, 89, 112, 164, 205, 294, 364, 505, 630, 845, 1052, 1393, 1719, 2235, 2762, 3533, 4343, 5506, 6730, 8443, 10296, 12786, 15531, 19161, 23161, 28374, 34201, 41621, 49975, 60513, 72385, 87200, 103999, 124670, 148209

Offset: 0

Omar E. Pol, Nov 29 2010

Here the "emergent parts" of the partitions of n are defined to be the parts (with multiplicity) of all the partitions that do not contain "1" as a part, removed by one copy of the smallest part of every partition. Note that these parts are located in the head of the last section of the set of partitions of n.
Also, here the "filler parts" of the partitions of n are defined to be the parts of the last section of the set of partitions of n that are not the emergent parts.
For n >= 4, length of row n of A183152. - Omar E. Pol, Aug 08 2011
Also total number of parts of the regions that do not contain 1 as a part in the last section of the set of partitions of n (cf. A083751, A187219). - Omar E. Pol, Mar 04 2012

			For n = 6 the partitions of 6 contain four "emergent" parts: (3), (4), (2), (2), so a(6) = 4. See below the location of the emergent parts.
6
(3) + 3
(4) + 2
(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
For a(10) = 22 see the link for the location of the 22 "emergent parts" (colored yellow and green) and the location of the 42 "filler parts" (colored blue) in the last section of the set of partitions of 10.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Omar E. Pol, Illustration: How to build the last section of the set of partitions (copy, paste and fill)
Omar E. Pol, Illustration of the shell model of partitions (2D view)

Cf. A000041, A002865, A135010, A138121, A138135, A182700, A182709, A182740.

b:= proc(n, i) option remember; local t, h;
      if n<0 then [0, 0, 0]
    elif n=0 then [0, 1, 0]
    elif i<2 then [0, 0, 0]
    else t:= b(n, i-1); h:= b(n-i, i);
         [t[1]+h[1]+h[2], t[2], t[3]+h[3]+h[1]]
      fi
    end:
a:= n-> b(n, n)[3]:
seq (a(n), n=0..50);  # Alois P. Heinz, Oct 21 2011

b[n_, i_] := b[n, i] = Module[{t, h}, Which[n<0, {0, 0, 0}, n == 0, {0, 1, 0}, i<2 , {0, 0, 0}, True, t = b[n, i-1]; h = b[n-i, i]; Join [t[[1]] + h[[1]] + h[[2]], t[[2]], t[[3]] + h[[3]] + h[[1]] ]]]; a[n_] := b[n, n][[3]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 18 2015, after Alois P. Heinz *)

a(n) = A138135(n) - A002865(n), n >= 1.
From Omar E. Pol, Oct 21 2011: (Start)
a(n) = A006128(n) - A006128(n-1) - A000041(n), n >= 1.
a(n) = A138137(n) - A000041(n), n >= 1. (End)
a(n) = A076276(n) - A006128(n-1), n >= 1. - Omar E. Pol, Oct 30 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

A195820 Total number of smallest parts in all partitions of n that do not contain 1 as a part.

Original entry on oeis.org

0, 1, 1, 3, 2, 7, 5, 12, 13, 22, 22, 43, 43, 67, 81, 117, 133, 195, 223, 312, 373, 492, 584, 782, 925, 1190, 1433, 1820, 2170, 2748, 3268, 4075, 4872, 5997, 7150, 8781, 10420, 12669, 15055, 18198, 21535, 25925, 30602, 36624, 43201, 51428, 60478, 71802, 84215

Offset: 1

Omar E. Pol, Oct 19 2011

nonn

Total number of smallest parts in all partitions of the head of the last section of the set of partitions of n.

			For n = 8 the seven partitions of 8 that do not contain 1 as a part are:
.  (8)
.  (4) + (4)
.   5  + (3)
.   6  + (2)
.   3  +  3  + (2)
.   4  + (2) + (2)
.  (2) + (2) + (2) + (2)
Note that in every partition the smallest parts are shown between parentheses. The total number of smallest parts is 1+2+1+1+1+2+4 = 12, so a(8) = 12.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews, The number of smallest parts in the partitions of n
A. Folsom and K. Ono, The spt-function of Andrews
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2020.
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13, arXiv:1011.1955 [math.NT], 2010.
K. Ono, Congruences for the Andrews spt-function
Wikipedia, Spt function

Cf. A000041, A000070, A002865, A092269, A135010, A138121, A138135, A138137, A182984.

b:= proc(n, i) option remember;
      `if`(n=0 or i<2, 0, b(n, i-1)+
       add(`if`(n=i*j, j, b(n-i*j, i-1)), j=1..n/i))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..60); # Alois P. Heinz, Apr 09 2012

Table[s = Select[IntegerPartitions[n], ! MemberQ[#, 1] &]; Plus @@ Table[Count[x, Min[x]], {x, s}], {n, 50}] (* T. D. Noe, Oct 19 2011 *)
b[n_, i_] := b[n, i] = If[n==0 || i<2, 0, b[n, i-1] + Sum[If[n== i*j, j, b[n-i*j, i-1]], {j, 1, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)

def A195820(n):
    return sum(list(p).count(min(p)) for p in Partitions(n,min_part=2))
# D. S. McNeil, Oct 19 2011

a(n) = A092269(n) - A000070(n-1).
G.f.: Sum_{i>=2} x^i/(1 - x^i) * Product_{j>=i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 03 2017
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) * (1 - (72 + 5*Pi^2)*sqrt(6) / (144*Pi*sqrt(n))). - Vaclav Kotesovec, Jul 31 2017

More terms from D. S. McNeil, Oct 19 2011

A207032 Triangle read by rows: T(n,k) = number of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 3, 0, 1, 7, 1, 2, 0, 1, 9, 6, 2, 2, 0, 1, 15, 4, 4, 1, 2, 0, 1, 19, 13, 4, 5, 1, 2, 0, 1, 32, 10, 10, 3, 4, 1, 2, 0, 1, 40, 24, 10, 9, 4, 4, 1, 2, 0, 1, 60, 23, 18, 8, 8, 3, 4, 1, 2, 0, 1, 78, 46, 22, 19, 8, 9, 3, 4, 1, 2, 0, 1

Offset: 1

Omar E. Pol, Feb 17 2012

For the calculation of row n, the number of odd/even parts, etc, take the row n from the triangle A207031 and then follow the same rules of A206563.

			Triangle begins:
  1;
  1,   1;
  3,   0,  1;
  3,   3,  0,  1;
  7,   1,  2,  0, 1;
  9,   6,  2,  2, 0, 1;
  15,  4,  4,  1, 2, 0, 1;
  19, 13,  4,  5, 1, 2, 0, 1;
  32, 10, 10,  3, 4, 1, 2, 0, 1;
  40, 24, 10,  9, 4, 4, 1, 2, 0, 1;
  60, 23, 18,  8, 8, 3, 4, 1, 2, 0, 1;
  78, 46, 22, 19, 8, 9, 3, 4, 1, 2, 0, 1;

Cf. A006128, A066897, A066898, A135010, A138121, A138135, A138137, A141285, A181187, A182703, A206563, A207031.

It appears that T(n,k) = abs(Sum_{j=k..n} (-1)^j*A207031(n,j)).

It appears that A182703(n,k) = T(n,k) - T(n,k+2). - Omar E. Pol, Feb 26 2012

A194548 Triangle read by rows: T(n,k) = number of parts in the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 2, 2, 1, 4, 3, 3, 2, 3, 2, 2, 1, 5, 4, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1, 5, 4, 4, 3, 4, 3, 3, 2, 3, 3, 2, 2, 2, 1, 6, 5, 5, 4, 4, 4, 3, 4, 3, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 2, 1, 6, 5, 5, 4, 5, 4, 4, 3, 4, 4, 3, 3, 3, 2, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1

Offset: 1

Omar E. Pol, Dec 11 2011

			Written as a triangle:
  0;
  1;
  1;
  2,1;
  2,1;
  3,2,2,1;
  3,2,2,1;
  4,3,3,2,2,2,1;
  4,3,3,2,3,2,2,1;
  5,4,4,3,3,3,2,3,2,2,2,1;
  5,4,4,3,4,3,3,2,3,3,2,2,2,1;
  6,5,5,4,4,4,3,4,3,3,3,2,4,3,3,2,3,2,2,2,1;
  6,5,5,4,5,4,4,3,4,4,3,3,3,2,4,3,3,3,2,3,2,2,2,1;

Alois P. Heinz, Rows n = 1..33, flattened
Tilman Piesk, Table for A194602, showing the non-one addends.

Row sums give A138135. Row n has length A187219(n).

Cf. A002865, A135010, A138121, A193173, A194546, A194547, A194549.

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      if n<2 then 0 else l:= NULL; b(n, 2, 0); l fi
    end:
seq(T(n), n=1..15);  # Alois P. Heinz, Dec 19 2011

T[n_] := Module[{b, l}, b[n0_, i_, t_] :=
     If[n0==0, l = Append[l, t],
     If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]];
     If[n<2, {0}, l = {}; b[n, 2, 0]; l]];
Table[T[n], {n, 1, 15}]  // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

More terms from Alois P. Heinz, Dec 19 2011

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 *)

A194448 Number of parts > 1 in the n-th region of the shell model of partitions.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 3, 1, 1, 14, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 1, 21, 1, 2, 1, 4, 1, 2

Offset: 1

Omar E. Pol, Dec 10 2011

Also triangle read by rows: T(n,k) = number of parts > 1 in the k-th region of the last section of the set of partitions of n.

			Written as a triangle:
0;
1;
1;
1,2;
1,2;
1,2,1,4;
1,2,1,4;
1,2,1,4,1,1,7;
1,2,1,4,1,2,1,8;
1,2,1,4,1,1,7,1,2,1,1,12;
1,2,1,4,1,2,1,8,1,1,3,1,1,14;
1,2,1,4,1,1,7,1,2,1,1,12,1,2,1,4,1,2,1,1,21;

Row sums give A138135. Row n has length A187219(n).

Cf. A000041, A002865, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A194447, A194449.

cp's OEIS Frontend

A138137 First differences of A006128.

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

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A207031 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the last section of the set of partitions of n.

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A182699 Number of emergent parts in all 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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A195820 Total number of smallest parts in all partitions of n that do not contain 1 as a part.

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A207032 Triangle read by rows: T(n,k) = number of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

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