A187219 -id:A187219 - cp's OEIS frontend

A186412 Sum of all parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7, 4, 25, 2, 6, 3, 13, 5, 4, 38, 3, 7, 4, 14, 3, 9, 5, 49, 2, 6, 3, 13, 5, 4, 23, 4, 10, 6, 5, 69, 3, 7, 4, 14, 3, 9, 5, 27, 5, 4, 15, 7, 6, 87, 2, 6, 3, 13, 5, 4, 23, 4, 10, 6, 5, 39, 3, 9, 5, 19, 4, 12, 7, 6, 123

Offset: 1

Omar E. Pol, Aug 12 2011

Also triangle read by rows: T(j,k) = sum of all parts in the k-th region of the last section of the set of partitions of j. See Example section. For more information see A135010. - Omar E. Pol, Nov 26 2011

For the definition of "region" see A206437. - Omar E. Pol, Aug 19 2013

			Contribution from Omar E. Pol, Nov 26 2011 (Start):
Written as a triangle:
1;
3;
5;
2,9;
3,12;
2,6,3,20;
3,7,4,25;
2,6,3,13,5,4,38;
3,7,4,14,3,9,5,49;
2,6,3,13,5,4,23,4,10,6,5,69;
3,7,4,14,3,9,5,27,5,4,15,7,6,87;
2,6,3,13,5,4,23,4,10,6,5,39,3,9,5,19,4,12,7,6,123;
(End)
From _Omar E. Pol_, Aug 18 2013: (Start)
Illustration of initial terms (first seven regions):
.                                             _ _ _ _ _
.                                     _ _ _  |_ _ _ _ _|
.                           _ _ _ _  |_ _ _|       |_ _|
.                     _ _  |_ _ _ _|                 |_|
.             _ _ _  |_ _|     |_ _|                 |_|
.       _ _  |_ _ _|             |_|                 |_|
.   _  |_ _|     |_|             |_|                 |_|
.  |_|   |_|     |_|             |_|                 |_|
.
.   1     3       5     2         9       3          12
.
(End)

Row sums of triangle A186114 and of triangle A193870.
Row j has length A187219(j).
Row sums give A138879.
Right border gives A046746, j >= 1.
Records give A046746, j >= 1.
Partial sums give A182244.
Cf. A000041, A002865, A066186, A135010, A138121, A138879, A194436, A194437, A194438, A194439, A194446, A194447.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
A186412 = {}; l = {};
For[j = 1, j <= 50, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[A186412, Total@Take[Reverse[First /@ lex[mx]], j - i]];
  ];
A186412  (* Robert Price, Jul 25 2020 *)

a(A000041(n)) = A046746(n).

A336812 Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n is constructed replacing every term of row n of A336811 with its divisors.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 20 2020

Here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the corresponce between all parts of the last section of the set of partitions of n and all divisors of all terms of the n-th row of A336811, with n >= 1. The mentionded parts and the mentioned divisors are the same numbers (see Example section).

For an equivalent table showing the same kind of correspondence for all partitions of all positive integers see the supersequence A338156.

			Triangle begins:
  [1];
  [1, 2];
  [1, 3],       [1];
  [1, 2, 4],    [1, 2],    [1];
  [1, 5],       [1, 3],    [1, 2], [1],    [1];
  [1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1];
  ...
For n = 6 the 6th row of A336811 is [6, 4, 3, 2, 2, 1, 1] so replacing every term with its divisors we have {[1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]} the same as the 6th row of this triangle.
Also, if the sequence is written as an irregular tetrahedron so the first six slices are:
  -------------
  [1],
  -------------
  [1, 2];
  -------------
  [1, 3],
  [1];
  -------------
  [1, 2, 4],
  [1, 2],
  [1];
  -------------
  [1, 5],
  [1, 3],
  [1, 2],
  [1],
  [1];
  -------------
  [1, 2, 3, 6],
  [1, 2, 4],
  [1, 3],
  [1, 2],
  [1, 2],
  [1],
  [1];
  -------------
The above slices appear in the lower zone of the following table which shows the correspondence between the mentioned divisors and the parts of the last section of the set of partitions of the positive integers.
The table is infinite. It is formed by three zones as follows:
The upper zone shows the last section of the set of partitions of every positive integer.
The lower zone shows the same numbers but arranged as divisors in accordance with the slices of the tetrahedron mentioned above.
Finally the middle zone shows the connection between the upper zone and the lower zone.
For every positive integer the numbers in the upper zone are the same numbers as in the lower zone.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| n |         |  1  |   2   |    3    |     4     |      5      |       6       |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   |         |     |       |         |           |             |  6            |
| P |         |     |       |         |           |             |  3 3          |
| A |         |     |       |         |           |             |  4 2          |
| R |         |     |       |         |           |             |  2 2 2        |
| T |         |     |       |         |           |  5          |    1          |
| I |         |     |       |         |           |  3 2        |      1        |
| T |         |     |       |         |  4        |    1        |      1        |
| I |         |     |       |         |  2 2      |      1      |        1      |
| O |         |     |       |  3      |    1      |      1      |        1      |
| N |         |     |  2    |    1    |      1    |        1    |          1    |
| S |         |  1  |    1  |      1  |        1  |          1  |            1  |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   | A207031 |  1  |  2 1  |  3 1 1  |  6 3 1 1  |  8 3 2 1 1  | 15 8 4 2 1 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |  |/|/|/|/|/|  |
| I | A182703 |  1  |  1 1  |  2 0 1  |  3 2 0 1  |  5 1 1 0 1  |  7 4 2 1 0 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |  * * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |  1 2 3 4 5 6  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |  = = = = = =  |
|   | A207383 |  1  |  1 2  |  2 0 3  |  3 4 0 4  |  5 2 3 0 5  |  7 8 6 4 0 6  |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |  1 2 3     6  |
| D |---------|-----|-------|---------|-----------|-------------|---------------|
| I | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
| V |---------|-----|-------|---------|-----------|-------------|---------------|
| I | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| S |---------|-----|-------|---------|-----------|-------------|---------------|
| O | A027750 |     |       |         |           |  1          |  1 2          |
| R | A027750 |     |       |         |           |  1          |  1 2          |
| S |---------|-----|-------|---------|-----------|-------------|---------------|
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
Note that every row in the lower zone lists A027750.
The "section" is the simpler substructure of the set of partitions of n that has this property in the three zones.
Also the lower zone for every positive integer can be constructed using the first n terms of A002865. For example: for n = 6 we consider the first 6 terms of A002865 (that is [1, 0, 1, 1, 2, 2]) and then the 6th slice is formed by a block with the divisors of 6, no block with the divisors of 5, one block with the divisors of 4, one block with the divisors of 3, two blocks with the divisors of 2 and two blocks with the divisors of 1.
Note that the lower zone is also in accordance with the tower (a polycube) described in A221529 in which its terraces are the symmetric representation of sigma starting from the top (cf. A237593) and the heights of the mentioned terraces are the partition numbers A000041 starting from the base.
The tower has the same volume (also the same number of cubes) equal to A066186(n) as a prism of partitions of size 1*n*A000041(n).
The above table shows the growth step by step of both the prism of partitions and its associated tower since the number of parts in the last section of the set of partitions of n is equal to A138137(n) equaling the number of divisors in the n-th slice of the lower table and equaling the same the number of terms in the n-th row of triangle. Also the sum of all parts in the last section of the set of partitions of n is equal to A138879(n) equaling the sum of all divisors in the n-th slice of the lower table and equaling the sum of the n-th row of triangle.

Paolo Xausa, Table of n, a(n) for n = 1..10206 (rows 1..23 of triangle, flattened)

Companion and subsequence of A338156.
Row lengths give A138137.
Row sums give A138879.
Cf. A000041, A000070, A002260, A002865, A006128, A024916, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A187219, A207031, A207038, A207383, A221529, A221530, A221531, A237593, A245095, A221649, A221650, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340035, A340061, A350357.

A336812[row_]:=Flatten[Table[ConstantArray[Divisors[row-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,row-1}]];
Array[A336812,10] (* Generates 10 rows *) (* Paolo Xausa, Feb 16 2023 *)

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

A182709 Sum of the emergent parts of the partitions of n.

Original entry on oeis.org

0, 0, 0, 2, 3, 11, 14, 33, 45, 81, 109, 185, 237, 372, 490, 715, 928, 1326, 1693, 2348, 2998, 4032, 5119, 6795, 8530, 11132, 13952, 17927, 22314, 28417, 35126, 44279, 54532, 68062, 83422, 103427, 126063, 155207, 188506, 230547, 278788, 339223, 408482

Offset: 1

Omar E. Pol, Nov 28 2010, 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. For more information see A182699.

Also total sum 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=7 the partitions of 7 that do not contain "1" as a part are
7
4 + 3
5 + 2
3 + 2 + 2
Then remove one copy of the smallest part of every partition. The rest are the emergent parts:
.,
4, .
5, .
3, 2, .
The sum of these parts is 4 + 5 + 3 + 2 = 14, so a(7)=14.
For n=10 the illustration in the link shows the location of the emergent parts (colored yellow and green) and the location of the filler parts (colored blue) in the last section of the set of partitions of 10.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jason Kimberley)
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, A135010, A138121, A138879, A138880, A182699, A182703, A182708, A182740, A182742, A182743.

Row sums of A183152.

b:= proc(n, i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<2 then 0
    else b(n, i-1) +b(n-i, i)
      fi
    end:
c:= proc(n, i, k) option remember;
      if n<0 then 0
    elif n=0 then k
    elif i<2 then 0
    else c(n, i-1, k) +c(n-i, i, i)
      fi
    end:
a:= n-> n*b(n, n) - c(n, n, 0):
seq(a(n), n=1..40);  #  Alois P. Heinz, Dec 01 2010

f[n_]:=Total[Flatten[Most/@Select[IntegerPartitions[n],!MemberQ[#,1]&]]]; Table[f[i],{i,50}] (* Harvey P. Dale, Dec 28 2010 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i<2, 0, True, b[n, i-1] + b[n - i, i]]; c[n_, i_, k_] := c[n, i, k] = Which[n<0, 0, n==0, k, i<2, 0, True, c[n, i-1, k] + c[n-i, i, i]]; a[n_] := n*b[n, n] - c[n, n, 0]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Oct 08 2015, after Alois P. Heinz *)

a(n) = A138880(n) - A182708(n).
a(n) = A066186(n) - A066186(n-1) - A046746(n) = A138879(n) - A046746(n). - Omar E. Pol, Aug 01 2013
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (12*sqrt(2*n)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n)). - Vaclav Kotesovec, Jan 03 2019, extended Jul 05 2019

More terms from Alois P. Heinz, Dec 01 2010

A211978 Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n.

Original entry on oeis.org

0, 2, 6, 12, 24, 40, 70, 108, 172, 256, 384, 550, 798, 1112, 1560, 2136, 2926, 3930, 5288, 6996, 9260, 12104, 15798, 20412, 26348, 33702, 43044, 54588, 69090, 86906, 109126, 136270, 169854, 210732, 260924, 321752, 396028, 485624, 594402, 725174, 883092, 1072208

Offset: 0

Omar E. Pol, Jan 03 2013

nonn

Also twice A006128, because the total number of parts in all partitions of n equals the sum of largest parts of all partitions of n. For a proof without words see the illustration of initial terms. Note that the sum of the lengths of all horizontal segments equals the sum of largest parts of all partitions of n. On the other hand, the sum of the lengths of all vertical segments equals the total number of parts of all partition of n. Therefore the sum of lengths of all horizontal segments equals the sum of lengths of all vertical segments.
a(n) is also the sum of the semiperimeters of the Ferrers boards of the partitions of n. Example: a(2)=6; indeed, the Ferrers boards of the partitions [2] and [1,1] of 2 are 2x1 rectangles; the sum of their semiperimeters is 3 + 3 = 6. - Emeric Deutsch, Oct 07 2016
a(n) is also the sum of the semiperimeters of the regions of the set of partitions of n. See the first illustration in the Example section. For more information see A278355. - Omar E. Pol, Nov 23 2016

			Illustration of initial terms as a minimalist diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                         _ _ _      |
.                                         _ _ _|_    |
.                                         _ _    |   |
.                             _ _ _ _ _   _ _|_ _|_  |
.                             _ _ _    |  _ _ _    | |
.                   _ _ _ _   _ _ _|_  |  _ _ _|_  | |
.                   _ _    |  _ _    | |  _ _    | | |
.           _ _ _   _ _|_  |  _ _|_  | |  _ _|_  | | |
.     _ _   _ _  |  _ _  | |  _ _  | | |  _ _  | | | |
. _   _  |  _  | |  _  | | |  _  | | | |  _  | | | | |
.  |   | |   | | |   | | | |   | | | | |   | | | | | |
.
. 2    6     12        24         40          70
.
Also using the elements from the diagram we can draw an infinite Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n) as shown below:
.
11...........................................................
.                                                           /\
.                                                          /  \
.                                                         /    \
7..................................                      /      \
.                                 /\                    /        \
5....................            /  \                /\/          \
.                   /\          /    \          /\  /              \
3..........        /  \        /      \        /  \/                \
2.....    /\      /    \    /\/        \      /                      \
1..  /\  /  \  /\/      \  /            \  /\/                        \
0 /\/  \/    \/          \/              \/                            \
. 0,2,  6,   12,         24,             40,                          70...
.

Michael De Vlieger, Table of n, a(n) for n = 0..3000
Toufik Mansour, Armend Sh. Shabani, Enumerations on bargraphs, Discrete Math. Lett. (2019) Vol. 2, 65-94.
Omar E. Pol, Visualization of regions in a diagram for A006128
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139582 is twice A000041. A220909 is twice A066186.

Cf. A006128, A135010, A141285, A186114, A193870, A187219, A194446, A194447, A206437, A211026, A220517, A225600, A278355.

Q := sum(x^j/(1-x^j), j = 1 .. i): R := product(1-x^j, j = 1 .. i): g := sum(x^i*(1+i+Q)/R, i = 1 .. 100): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 41); # Emeric Deutsch, Oct 07 2016

Array[2 Sum[DivisorSigma[0, m] PartitionsP[# - m], {m, #}] &, 42, 0] (* Michael De Vlieger, Mar 20 2020 *)

a(n) = 2*A006128(n).
a(n) = A225600(2*A000041(n)) = A225600(A139582(n)), n >= 1.
a(n) = (Sum_{m=1..p(n)} A194446(m)) + (Sum_{m=1..p(n)} A141285(m)) = 2*Sum_{m=1..p(n)} A194446(m) = 2*Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1.
The trivariate g.f. G(t,s,x) of the partitions of a nonnegative integer relative to weight (marked by x), number of parts (marked by t), and largest part (marked by s) is G(t,s,x) = Sum_{i>=1} t*s^i*x^i/product_{j=1..i} (1-tx^j). Setting s = t, we obtain the bivariate g.f. of the partitions relative to weight (marked by x) and semiperimeter of the Ferrers board (marked by t). The g.f. of a(n) is g(x) = Sum_{i>=1} ((x^i*(1 + i + Q(x))/R(x)), where Q(x) = sum_{j=1..i} (x^j/(1 - x^j)) and R(x) = product_{j=1..i}(1-x^j). g(x) has been obtained by setting t = 1 in dG(t,t,x))/dt. - Emeric Deutsch, Oct 07 2016

A225600 Toothpick sequence related to integer partitions (see Comments lines for definition).

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 14, 15, 19, 24, 27, 28, 33, 40, 42, 43, 47, 49, 52, 53, 59, 70, 73, 74, 79, 81, 85, 86, 93, 108, 110, 111, 115, 117, 120, 121, 127, 131, 136, 137, 141, 142, 150, 172, 175, 176, 181, 183, 187, 188, 195, 199, 202, 203, 209, 211, 216, 217, 226, 256

Offset: 0

Omar E. Pol, Jul 28 2013

nonn

This infinite toothpick structure is a minimalist diagram of regions of the set of partitions of all positive integers. For the definition of "region" see A206437. The sequence shows the growth of the diagram as a cellular automaton in which the "input" is A141285 and the "output” is A194446.
To define the sequence we use the following rules:
We start in the first quadrant of the square grid with no toothpicks.
If n is odd we place A141285((n+1)/2) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the grid point (0, (n+1)/2).
If n is even we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. In this case the number of toothpicks added in vertical direction is equal to A194446(n/2).
The sequence gives the number of toothpicks after n stages. A220517 (the first differences) gives the number of toothpicks added at the n-th stage.
Also the toothpick structure (HV/HHVV/HHHVVV/HHV/HHHHVVVVV...) can be transformed in a Dyck path (UDUUDDUUUDDDUUDUUUUDDDDD...) in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps, so the sequence can be represented by the vertices (or the number of steps from the origin) of the Dyck path. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See Example section. See also A211978, A220517, A225610.

			For n = 30 the structure has 108 toothpicks, so a(30) = 108.
.                               Diagram of regions
Partitions of 7                 and partitions of 7
.                                   _ _ _ _ _ _ _
7                               15  _ _ _ _      |
4 + 3                               _ _ _ _|_    |
5 + 2                               _ _ _    |   |
3 + 2 + 2                           _ _ _|_ _|_  |
6 + 1                           11  _ _ _      | |
3 + 3 + 1                           _ _ _|_    | |
4 + 2 + 1                           _ _    |   | |
2 + 2 + 2 + 1                       _ _|_ _|_  | |
5 + 1 + 1                        7  _ _ _    | | |
3 + 2 + 1 + 1                       _ _ _|_  | | |
4 + 1 + 1 + 1                    5  _ _    | | | |
2 + 2 + 1 + 1 + 1                   _ _|_  | | | |
3 + 1 + 1 + 1 + 1                3  _ _  | | | | |
2 + 1 + 1 + 1 + 1 + 1            2  _  | | | | | |
1 + 1 + 1 + 1 + 1 + 1 + 1        1   | | | | | | |
.
.                                   1 2 3 4 5 6 7
.
Illustration of initial terms:
.
.                              _ _ _    _ _ _
.                _ _   _ _     _ _      _ _  |
.      _    _    _     _  |    _  |     _  | |
.            |    |     | |     | |      | | |
.
.      1    2     4     6       9        12
.
.
.                          _ _ _ _     _ _ _ _
.      _ _       _ _       _ _         _ _    |
.      _ _ _     _ _|_     _ _|_       _ _|_  |
.      _ _  |    _ _  |    _ _  |      _ _  | |
.      _  | |    _  | |    _  | |      _  | | |
.       | | |     | | |     | | |       | | | |
.
.        14        15         19          24
.
.
.                          _ _ _ _ _    _ _ _ _ _
.    _ _ _      _ _ _      _ _ _        _ _ _    |
.    _ _ _ _    _ _ _|_    _ _ _|_      _ _ _|_  |
.    _ _    |   _ _    |   _ _    |     _ _    | |
.    _ _|_  |   _ _|_  |   _ _|_  |     _ _|_  | |
.    _ _  | |   _ _  | |   _ _  | |     _ _  | | |
.    _  | | |   _  | | |   _  | | |     _  | | | |
.     | | | |    | | | |    | | | |      | | | | |
.
.       27         28         33            40
.
Illustration of initial terms as vertices (or the number of steps from the origin) of a Dyck path:
.
7                                    33
.                                    /\
5                      19           /  \
.                      /\          /    \
3            9        /  \     27 /      \
2       4    /\   14 /    \    /\/        \
1    1  /\  /  \  /\/      \  / 28         \
.    /\/  \/    \/ 15       \/              \
.   0  2   6    12          24              40
.

Omar E. Pol, Visualization of regions in a diagram for A006128
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000041, A006128, A135010, A138137, A139250, A139582, A141285, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225610.

a(A139582(n)) = a(2*A000041(n)) = 2*A006128(n) = A211978(n), n >= 1.

A339278 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which the partition number A000041(n-1) is the length of row n and every column k is A000203, the sum of divisors function.

Original entry on oeis.org

1, 3, 4, 1, 7, 3, 1, 6, 4, 3, 1, 1, 12, 7, 4, 3, 3, 1, 1, 8, 6, 7, 4, 4, 3, 3, 1, 1, 1, 1, 15, 12, 6, 7, 7, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 13, 8, 12, 6, 6, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 18, 15, 8, 12, 12, 6, 6, 7, 7, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Nov 29 2020

The sum of row n equals A138879(n), the sum of all parts in the last section of the set of partitions of n.

T(n,k) is also the number of cubic cells (or cubes) added at the n-th stage in the k-th level starting from the base in the tower described in A221529, assuming that the tower is an object under construction (see the example). - Omar E. Pol, Jan 20 2022

			Triangle begins:
   1;
   3;
   4,  1;
   7,  3,  1;
   6,  4,  3, 1, 1;
  12,  7,  4, 3, 3, 1, 1;
   8,  6,  7, 4, 4, 3, 3, 1, 1, 1, 1;
  15, 12,  6, 7, 7, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1;
  13,  8, 12, 6, 6, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
...
From _Omar E. Pol_, Jan 13 2022: (Start)
Illustration of the first six rows of triangle showing the growth of the symmetric tower described in A221529:
    Level k: 1              2         3        4       5      6     7
Stage
  n   _ _ _ _ _ _ _ _
     |            _  |
  1  |           |_| |
     |_ _ _ _ _ _ _ _|
     |          _    |
     |         | |_  |
  2  |         |_ _| |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _
     |        _      |        _  |
     |       | |     |       |_| |
  3  |       |_|_ _  |           |
     |         |_ _| |           |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _
     |      _        |      _    |      _  |
     |     | |       |     | |_  |     |_| |
  4  |     | |_      |     |_ _| |         |
     |     |_  |_ _  |           |         |
     |       |_ _ _| |           |         |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _ _ _ _ _
     |    _          |    _      |    _    |    _  |    _  |
     |   | |         |   | |     |   | |_  |   |_| |   |_| |
     |   | |         |   |_|_ _  |   |_ _| |       |       |
  5  |   |_|_        |     |_ _| |         |       |       |
     |       |_ _ _  |           |         |       |       |
     |       |_ _ _| |           |         |       |       |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _ _ _ _
     |  _            |  _        |  _      |  _    |  _    |  _  |  _  |
     | | |           | | |       | | |     | | |_  | | |_  | |_| | |_| |
     | | |           | | |_      | |_|_ _  | |_ _| | |_ _| |     |     |
     | | |_ _        | |_  |_ _  |   |_ _| |       |       |     |     |
  6  | |_    |       |   |_ _ _| |         |       |       |     |     |
     |   |_  |_ _ _  |           |         |       |       |     |     |
     |     |_ _ _ _| |           |         |       |       |     |     |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _|_ _ _|
.
Every cell in the diagram of the symmetric representation of sigma represents a cubic cell or cube.
For n = 6 and k = 3 we add four cubes at 6th stage in the third level of the structure of the tower starting from the base so T(6,3) = 4.
For n = 9 another connection with the tower is as follows:
First we take the columns from the above triangle and build a new triangle in which all columns start at row 1 as shown below:
.
   1,  1,  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
   3,  3,  3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3;
   4,  4,  4, 4, 4, 4, 4, 4, 4, 4, 4;
   7,  7,  7, 7, 7, 7, 7;
   6,  6,  6, 6, 6;
  12, 12, 12;
   8,  8;
  15;
  13;
.
Then we rotate the triangle by 90 degrees as shown below:
                                       _
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  |_|_
  1, 3;                               |   |
  1, 3;                               |   |
  1, 3;                               |   |
  1, 3;                               |_ _|_
  1, 3, 4;                            |   | |
  1, 3, 4;                            |   | |
  1, 3, 4;                            |   | |
  1, 3, 4;                            |_ _|_|_
  1, 3, 4, 7;                         |     | |
  1, 3, 4, 7;                         |_ _ _| |_
  1, 3, 4, 7, 6;                      |     |   |
  1, 3, 4, 7, 6;                      |_ _ _|_ _|_
  1, 3, 4, 7, 6, 12;                  |_ _ _ _| | |_
  1, 3, 4, 7, 6, 12, 8;               |_ _ _ _|_|_ _|_ _
  1, 3, 4, 7, 6, 12, 8, 15; 13;       |_ _ _ _ _|_ _|_ _|
.
                                         Lateral view
                                         of the tower
.                                      _ _ _ _ _ _ _ _ _
                                      |_| | | | | | |   |
                                      |_ _|_| | | | |   |
                                      |_ _|  _|_| | |   |
                                      |_ _ _|    _|_|   |
                                      |_ _ _|  _|    _ _|
                                      |_ _ _ _|     |
                                      |_ _ _ _|  _ _|
                                      |         |
                                      |_ _ _ _ _|
.
                                           Top view
                                         of the tower
.
The sum of the m-th row of the new triangle equals A024916(j) where j is the length of the m-th row, equaling the number of cubic cells in the m-th level of the tower. For example: the last row of triangle has 9 terms and the sum of the last row is 1 + 3 + 4 + 7 + 6 + 12 + 8 + 15 + 13 = A024916(9) = 69, equaling the number of cubes in the base of the tower. (End)

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of triangle, flattened)

Sum of divisors of A336811.
Row n has length A000041(n-1).
Every column gives A000203.
The length of the m-th block in row n is A187219(m), m >= 1.
Row sums give A138879.
Cf. A337209 (another version).
Cf. A272172 (analog for the stepped pyramid described in A245092).
Cf. A024916, A135010, A138121, A138137, A138785, A221529, A235791, A236104, A237270, A237591, A237593, A238442, A338156, A340423, A345023.

A339278[rowmax_]:=Table[Flatten[Table[ConstantArray[DivisorSigma[1,n-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]],{n,rowmax}];
A339278[15] (* Generates 15 rows *) (* Paolo Xausa, Feb 17 2023 *)

f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (sigma(n))); my(s=0); while (k <= f(n-1), s++; n--;); sigma(1+s);}
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ Michel Marcus, Jan 13 2021

A339278(rowmax)=vector(rowmax,n,concat(vector(n,m,vector(numbpart(m-1)-numbpart(m-2),i,sigma(n-m+1)))));
A339278(15) \\ Generates 15 rows \\ Paolo Xausa, Feb 17 2023

a(m) = A000203(A336811(m)).

T(n,k) = A000203(A336811(n,k)).

A083751 Number of partitions of n into >= 2 parts and with minimum part >= 2.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 6, 7, 11, 13, 20, 23, 33, 40, 54, 65, 87, 104, 136, 164, 209, 252, 319, 382, 477, 573, 707, 846, 1038, 1237, 1506, 1793, 2166, 2572, 3093, 3659, 4377, 5169, 6152, 7244, 8590, 10086, 11913, 13958, 16423, 19195, 22518, 26251, 30700, 35716

Offset: 1

Jon Perry, Jun 17 2003

nonn

Also number of partitions of n such that the largest part is at least 2 and occurs at least twice. Example: a(6)=3 because we have [3,3],[2,2,2] and [2,2,1,1]. - Emeric Deutsch, Mar 29 2006
Also number of partitions of n that contain emergent parts (Cf. A182699). - Omar E. Pol, Oct 21 2011
Also number of regions in the last section of the set of partitions of n that do not contain 1 as a part (cf. A187219). - Omar E. Pol, Mar 04 2012
Schneider calls these "nuclear partitions" and gives a remarkable formula relating a(n), the number of partitions of n, and a sum over the two greatest parts of each such partition. - Charles R Greathouse IV, Dec 04 2019

			a(6) = 3, as 6 = 2+4 = 3+3 = 2+2+2.
a(6) = 3 because 6 = 2+4 = 3+3 = 2+2+2.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Robert Schneider, Nuclear partitions and a formula for p(n), arXiv:1912.00575 [math.NT], 2019.

Cf. A053445, A072380, A008483, A026796, A035989, A036000, A002865, A081094.

First differences of A000094.

g:=sum(x^(2*j)/product(1-x^i,i=1..j),j=2..50): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..51); # Emeric Deutsch, Mar 29 2006

Drop[CoefficientList[Series[1/Product[(1-x^k)^1, {k, 2, 50}], {x, 0, 50}], x]-1, 2]
(* or *) Table[Count[IntegerPartitions[n], q_List /; Length[q] > 1 && Min[q] >= 2 ], {n, 24}]

a(n) = A000041(n) - A000041(n-1) - 1, n > 1. - Vladeta Jovovic, Jun 18 2003
G.f.: Sum_{j>=2} x^(2j)/Product_{i=1..j} (1-x^i). - Emeric Deutsch, Mar 29 2006
a(n) = A002865(n) - 1, n > 1. - Omar E. Pol, Oct 21 2011
a(n) = A187219(n) - 1. - Omar E. Pol, Mar 04 2012

More terms from Vladeta Jovovic and Wouter Meeussen, Jun 18 2003

Description corrected by James Sellers, Jun 21 2003

A220517 First differences of A225600. Also A141285 and A194446 interleaved.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 2, 1, 4, 5, 3, 1, 5, 7, 2, 1, 4, 2, 3, 1, 6, 11, 3, 1, 5, 2, 4, 1, 7, 15, 2, 1, 4, 2, 3, 1, 6, 4, 5, 1, 4, 1, 8, 22, 3, 1, 5, 2, 4, 1, 7, 4, 3, 1, 6, 2, 5, 1, 9, 30, 2, 1, 4, 2, 3, 1, 6, 4, 5, 1, 4, 1, 8, 7, 4, 1, 7, 2, 6, 1, 5, 1, 10, 42

Offset: 1

Omar E. Pol, Feb 07 2013

Number of toothpicks added at n-th stage to the toothpick structure (related to integer partitions) of A225600.

			Written as an irregular triangle in which row n has length 2*A187219(n) we can see that the right border gives A000041 and the previous term of the last term in row n is n.
1,1;
2,2;
3,3;
2,1,4,5;
3,1,5,7;
2,1,4,2,3,1,6,11;
3,1,5,2,4,1,7,15;
2,1,4,2,3,1,6,4,5,1,4,1,8,22;
3,1,5,2,4,1,7,4,3,1,6,2,5,1,9,30;
2,1,4,2,3,1,6,4,5,1,4,1,8,7,4,1,7,2,6,1,5,1,10,42;
.
Illustration of the first seven rows of triangle as a minimalist diagram of regions of the set of partitions of 7:
.      _ _ _ _ _ _ _
. 15   _ _ _ _      |
.      _ _ _ _|_    |
.      _ _ _    |   |
.      _ _ _|_ _|_  |
. 11   _ _ _      | |
.      _ _ _|_    | |
.      _ _    |   | |
.      _ _|_ _|_  | |
.  7   _ _ _    | | |
.      _ _ _|_  | | |
.  5   _ _    | | | |
.      _ _|_  | | | |
.  3   _ _  | | | | |
.  2   _  | | | | | |
.  1    | | | | | | |
.
.      1 2 3 4 5 6 7
.
Also using the elements of this diagram we can draw a Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See below:
.
7..................................
.                                 /\
5....................            /  \                /\
.                   /\          /    \          /\  /
3..........        /  \        /      \        /  \/
2.....    /\      /    \    /\/        \      /
1..  /\  /  \  /\/      \  /            \  /\/
0 /\/  \/    \/          \/              \/
. 0,2,  6,   12,         24,             40... = A211978
.  1, 4,   9,       19,           33... = A179862
.

Omar E. Pol, Visualization of regions in a minimalist diagram for A006128
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000041, A006128, A135010, A138137, A141285, A179862, A186114, A186412, A187219, A194446, A206437, A211978, A220517, A225600, A225610.

a(2n-1) = A141285(n); a(2n) = A194446(n), n >= 1

A225610 Total number of parts in all partitions of n plus the sum of largest parts in all partitions of n plus the number of partitions of n plus n.

Original entry on oeis.org

1, 4, 10, 18, 33, 52, 87, 130, 202, 295, 436, 617, 887, 1226, 1709, 2327, 3173, 4244, 5691, 7505, 9907, 12917, 16822, 21690, 27947, 35685, 45506, 57625, 72836, 91500, 114760, 143143, 178235, 220908, 273268, 336670, 414041, 507298, 620455, 756398, 920470

Offset: 0

Omar E. Pol, Jul 29 2013

nonn

a(n) is also the total number of toothpicks in a toothpick structure which represents a diagram of regions of the set of partitions of n, n >= 1. The number of horizontal toothpicks is A225596(n). The number of vertical toothpicks is A093694(n). The difference between vertical toothpicks and horizontal toothpicks is A000041(n) - n = A000094(n+1). The total area (or total number of cells) of the diagram is A066186(n). The number of parts in the k-th region is A194446(k). The area (or number of cells) of the k-th region is A186412(k). For the definition of "region" see A206437. For a minimalist version of the diagram (which can be transformed into a Dyck path) see A211978. See also A225600.

			For n = 7 the total number of parts in all partitions of 7 plus the sum of largest parts in all partitions of 7 plus the number of partitions of 7 plus 7 is equal to A006128(7) + A006128(7) + A000041(7) + 7 = 54 + 54 + 15 + 7 = 130. On the other hand the number of toothpicks in the diagram of regions of the set of partitions of 7 is equal to 130, so a(7) = 130.
.                               Diagram of regions
Partitions of 7                 and partitions of 7
.                                   _ _ _ _ _ _ _
7                               15 |_ _ _ _      |
4 + 3                              |_ _ _ _|_    |
5 + 2                              |_ _ _    |   |
3 + 2 + 2                          |_ _ _|_ _|_  |
6 + 1                           11 |_ _ _      | |
3 + 3 + 1                          |_ _ _|_    | |
4 + 2 + 1                          |_ _    |   | |
2 + 2 + 2 + 1                      |_ _|_ _|_  | |
5 + 1 + 1                        7 |_ _ _    | | |
3 + 2 + 1 + 1                      |_ _ _|_  | | |
4 + 1 + 1 + 1                    5 |_ _    | | | |
2 + 2 + 1 + 1 + 1                  |_ _|_  | | | |
3 + 1 + 1 + 1 + 1                3 |_ _  | | | | |
2 + 1 + 1 + 1 + 1 + 1            2 |_  | | | | | |
1 + 1 + 1 + 1 + 1 + 1 + 1        1 |_|_|_|_|_|_|_|
.
.                                   1 2 3 4 5 6 7
.
Illustration of initial terms as the number of toothpicks in a diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                        |_ _ _      |
.                                        |_ _ _|_    |
.                                        |_ _    |   |
.                             _ _ _ _ _  |_ _|_ _|_  |
.                            |_ _ _    | |_ _ _    | |
.                   _ _ _ _  |_ _ _|_  | |_ _ _|_  | |
.                  |_ _    | |_ _    | | |_ _    | | |
.           _ _ _  |_ _|_  | |_ _|_  | | |_ _|_  | | |
.     _ _  |_ _  | |_ _  | | |_ _  | | | |_ _  | | | |
. _  |_  | |_  | | |_  | | | |_  | | | | |_  | | | | |
.|_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
.
. 4    10     18       33         52          87

Omar E. Pol, Illustration of the seven regions of 5
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000041, A000094, A006128, A066186, A093694, A133041, A135010, A138137, A139250, A139582, A141285, A182377, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225596, A225600.

a(n) = 2*A006128(n) + A000041(n) + n = A211978(n) + A133041(n) = A093694(n) + A006128(n) + n = A093694(n) + A225596(n).

cp's OEIS Frontend

A186412 Sum of all parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

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A336812 Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n is constructed replacing every term of row n of A336811 with its divisors.

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A182699 Number of emergent parts in all partitions of n.

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A182709 Sum of the emergent parts of the partitions of n.

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A211978 Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n.

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A225600 Toothpick sequence related to integer partitions (see Comments lines for definition).

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A339278 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which the partition number A000041(n-1) is the length of row n and every column k is A000203, the sum of divisors function.

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