A066633 -id:A066633 - cp's OEIS frontend

A340011 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the j-th row of triangle A127093 but with every term multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

1, 1, 2, 1, 1, 0, 3, 1, 2, 2, 1, 2, 0, 4, 1, 0, 3, 2, 4, 3, 1, 0, 0, 0, 5, 1, 2, 0, 4, 2, 0, 6, 3, 6, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 5, 2, 4, 0, 8, 3, 0, 9, 5, 10, 7, 1, 0, 0, 0, 0, 0, 7, 1, 2, 3, 0, 0, 6, 2, 0, 0, 0, 10, 3, 6, 0, 12, 5, 0, 15, 7, 14, 11, 1, 2, 0, 4, 0, 0, 0, 8

Offset: 1

Omar E. Pol, Dec 26 2020

This triangle is a condensed version of the more irregular triangle A340031.

For further information about the correspondence divisor/part see A338156.

			Triangle begins:
[1];
[1, 2],          [1];
[1, 0, 3],       [1, 2],       [2];
[1, 2, 0, 4],    [1, 0, 3],    [2, 4],    [3];
[1, 0, 0, 0, 5], [1, 2, 0, 4], [2, 0, 6], [3, 6], [5];
[...
Row sums give A066186.
Written as an irregular tetrahedron the first five slices are:
--
1;
-----
1, 2,
1;
--------
1, 0, 3,
1, 2,
2;
-----------
1, 2, 0, 4,
1, 0, 3,
2, 4,
3;
--------------
1, 0, 0, 0, 5,
1, 2, 0, 4,
2, 0, 6,
3, 6,
5;
--------------
Row sums give A339106.
The following table formed by four zones shows the correspondence between divisor and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |     |       |         |           |             |
| A |         |     |       |         |           |             |
| R |         |     |       |         |           |             |
| T |         |     |       |         |           |  5          |
| I |         |     |       |         |           |  3 2        |
| T |         |     |       |         |  4        |  4 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O | A127093 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
| C | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
| O |    -    |     |       |  2      |  2 4      |  2 0 6      |
| N |    -    |     |       |         |  3        |  3 6        |
| D |    -    |     |       |         |           |  5          |
|---|---------|-----|-------|---------|-----------|-------------|
.
This lower zone of the table is a condensed version of the "divisors" zone.

Paolo Xausa, Table of n, a(n) for n = 1..11480 (rows 1..40 of the triangle, flattened)

Row sums give A066186.
Nonzero terms give A340056.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340032, A340035, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340011row[n_]:=Flatten[Table[A127093row[n-m+1]PartitionsP[m-1],{m,n}]];
Array[A340011row,10] (* Paolo Xausa, Sep 28 2023 *)

A340032 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the row m of triangle A127093, with 1 <= m <= n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 26 2020

For further information about the correspondence divisor/part see A338156.

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 1, 2, 1, 0, 3;
  1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 2, 0, 4;
  1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 3, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5;
  ...
Written as an irregular tetrahedron the first five slices are:
  1;
  --
  1,
  1, 2;
  -----
  1,
  1,
  1, 2,
  1, 0, 3;
  --------
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 0, 3,
  1, 2, 0, 4;
  -----------
  1,
  1,
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 2,
  1, 0, 3,
  1, 0, 3,
  1, 2, 0, 4,
  1, 0, 0, 0, 5;
  --------------
  ...
The slices of the tetrahedron appear in the upper zone of the following table (formed by three zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
| D | A127093 |     |       |         |           |  1          |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A127093 |     |       |         |  1        |  1 2        |
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| S | A127093 |     |       |  1      |  1 2      |  1 0 3      |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
| L | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
| I |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| N | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| K |         |  |  |  |\|  |  |\|\|  |  |\|\|\|  |  |\|\|\|\|  |
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
| A |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| R |         |     |       |  3      |  3 1      |  3 1 1      |
| T |         |     |       |         |  2 2      |  2 2 1      |
| I |         |     |       |         |  4        |  4 1        |
| T |         |     |       |         |           |  3 2        |
| I |         |     |       |         |           |  5          |
| O |         |     |       |         |           |             |
| N |         |     |       |         |           |             |
| S |         |     |       |         |           |             |
|---|---------|-----|-------|---------|-----------|-------------|
.
The table is essentially the same table of A340035 but here, in the upper zone, every row is A127093 instead of A027750.
Also the above table is the table of A340031 upside down.

Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)

Row sums give A066186.
Nonzero terms gives A340035.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340032row[n_]:=Flatten[Table[ConstantArray[A127093row[m],PartitionsP[n-m]],{m,n}]];
Array[A340032row,7] (* Paolo Xausa, Sep 28 2023 *)

A340056 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the divisors of j multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 4, 1, 3, 2, 4, 3, 1, 5, 1, 2, 4, 2, 6, 3, 6, 5, 1, 2, 3, 6, 1, 5, 2, 4, 8, 3, 9, 5, 10, 7, 1, 7, 1, 2, 3, 6, 2, 10, 3, 6, 12, 5, 15, 7, 14, 11, 1, 2, 4, 8, 1, 7, 2, 4, 6, 12, 3, 15, 5, 10, 20, 7, 21, 11, 22, 15, 1, 3, 9, 1, 2, 4, 8, 2, 14, 3, 6, 9, 18, 5

Offset: 1

Omar E. Pol, Dec 27 2020

This triangle is a condensed version of the more irregular triangle A338156 which is the main sequence with further information about the correspondence divisor/part.

			Triangle begins:
  [1];
  [1, 2],    [1];
  [1, 3],    [1, 2],    [2];
  [1, 2, 4], [1, 3],    [2, 4], [3];
  [1, 5],    [1, 2, 4], [2, 6], [3, 6], [5];
  [...
The row sums of triangle give A066186.
Written as an irregular tetrahedron the first five slices are:
  1;
  -----
  1, 2,
  1;
  -----
  1, 3,
  1, 2,
  2;
  --------
  1, 2, 4,
  1, 3,
  2, 4,
  3;
  --------
  1, 5,
  1, 2, 4,
  2, 6,
  3, 6,
  5;
  --------
The row sums of tetrahedron give A339106.
The slices of the tetrahedron appear in the following table formed by four zones shows the correspondence between divisor and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |     |       |         |           |             |
| A |         |     |       |         |           |             |
| R |         |     |       |         |           |             |
| T |         |     |       |         |           |  5          |
| I |         |     |       |         |           |  3 2        |
| T |         |     |       |         |  4        |  4 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A027750 |     |       |  1      |  1 2      |  1   3      |
| I | A027750 |     |       |  1      |  1 2      |  1   3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A027750 |     |       |         |  1        |  1 2        |
| S | A027750 |     |       |         |  1        |  1 2        |
| O | A027750 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
| C | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
| O |    -    |     |       |  2      |  2 4      |  2   6      |
| N |    -    |     |       |         |  3        |  3 6        |
| D |    -    |     |       |         |           |  5          |
|---|---------|-----|-------|---------|-----------|-------------|
.
The lower zone is a condensed version of the "divisors" zone.

Paolo Xausa, Table of n, a(n) for n = 1..11528 (rows 1..75 of the triangle, flattened)

Nonzero terms of A340011.
Row sums give A066186.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340061.

A340056row[n_]:=Flatten[Table[Divisors[n-m]PartitionsP[m],{m,0,n-1}]];Array[A340056row,10] (* Paolo Xausa, Sep 01 2023 *)

A015716 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, one of which is k (1<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 3, 3, 3, 2, 2, 2, 1, 1, 1, 5, 4, 4, 3, 2, 2, 2, 1, 1, 1, 5, 5, 4, 3, 3, 3, 2, 2, 1, 1, 1, 7, 6, 5, 5, 4, 3, 3, 2, 2, 1, 1, 1, 8, 7, 6, 6, 4, 4, 4, 3, 2, 2, 1, 1

Offset: 1

Clark Kimberling

Row sums yield A015723. T(n,1)=A025147(n-1); T(n,2)=A015744(n-2); T(n,3)=A015745(n-3); T(n,4)=A015746(n-4); T(n,5)=A015750(n-5). - Emeric Deutsch, Mar 29 2006

Number of parts of size k in all partitions of n into distinct parts. Number of partitions of n-k into distinct parts not including a part of size k. - Franklin T. Adams-Watters, Jan 24 2012

			T(8,3)=2 because we have [5,3] and [4,3,1].
Triangle begins:
n/k 1 2 3 4 5 6 7 8 9 10
01: 1
02: 0 1
03: 1 1 1
04: 1 0 1 1
05: 1 1 1 1 1
06: 2 2 1 1 1 1
07: 2 2 1 2 1 1 1
08: 3 2 2 1 2 1 1 1
09: 3 3 3 2 2 2 1 1 1
10: 5 4 4 3 2 2 2 1 1 1
...
The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), which is row n = 6. - _Gus Wiseman_, May 07 2019

Mircea Merca, Table of n, a(n) for n = 1..7260

Cf. A015723, A015744, A015745, A015746, A015750, A025147, A027293, A066633.

Cf. A006128, A022629, A066189, A246867, A325504, A325506, A325513.

g:=product(1+x^j,j=1..50)*sum(t^i*x^i/(1+x^i),i=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 14 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Mar 29 2006
seq(seq(coeff(x^k*(product(1+x^j, j=1..n))/(1+x^k), x, n), k=1..n), n=1..13); # Mircea Merca, Feb 28 2014

z = 15; d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]; p[n_, k_] := p[n, k] = d[n][[k]]; s[n_] := s[n] = Flatten[Table[p[n, k], {k, 1, PartitionsQ[n]}]]; t[n_, k_] := Count[s[n], k]; u = Table[t[n, k], {n, 1, z}, {k, 1, n}]; TableForm[u] (* A015716 as a triangle *)
v = Flatten[u] (* A015716 as a sequence *)
(* Clark Kimberling, Mar 14 2014 *)

G.f.: G(t,x) = Product_{j>=1} (1+x^j) * Sum_{i>=1} t^i*x^i/(1+x^i). - Emeric Deutsch, Mar 29 2006
From Mircea Merca, Feb 28 2014: (Start)
a(n) = A238450(n) + A238451(n).
T(n,k) = Sum_{j=1..floor(n/k)} (-1)^(j-1)*A000009(n-j*k).
G.f.: for column k: q^k/(1+q^k)*(-q;q)_{inf}. (End)

A210947 Triangle read by rows: T(n,k) = total number of parts <= k of all partitions of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 12, 16, 18, 19, 20, 19, 27, 31, 33, 34, 35, 30, 41, 47, 50, 52, 53, 54, 45, 64, 73, 79, 82, 84, 85, 86, 67, 93, 108, 116, 121, 124, 126, 127, 128, 97, 138, 159, 172, 180, 185, 188, 190, 191, 192

Offset: 1

Omar E. Pol, May 01 2012

Row n lists the partial sums of row n of triangle A066633.

			Triangle begins:
1;
2,   3;
4,   5,  6;
7,  10,  11,  12;
12, 16,  18,  19,  20;
19, 27,  31,  33,  34,  35;
30, 41,  47,  50,  52,  53,  54;
45, 64,  73,  79,  82,  84,  85,  86;
67, 93, 108, 116, 121, 124, 126, 127, 128;

Alois P. Heinz, Rows n = 1..141, flattened

Column 1 is A000070(n-1). Right border gives A006128.

Cf. A066633, A181187, A206563, A210948, A210955.

p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local f, g;
      if n=0 then [1]
    elif i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));
         p (p (f, g), [0$i, g[1]])
      fi
    end:
T:= proc(n, k) option remember;
       b(n, n)[k+1] +`if`(k<2, 0, T(n, k-1))
    end:
seq (seq (T(n,k), k=1..n), n=1..11); # Alois P. Heinz, May 02 2012

p[f_, g_] := With[{m = Max[Length[f], Length[g]]}, PadRight[f, m, 0] + PadRight[g, m, 0]]; b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0, {1}, If[i == 1, {1, n}, f = b[n, i-1]; g = If[i>n, {0}, b[n-i, i]]; p[p[f, g], Append[Array[0&, i], g[[1]] ]]]]]; T[n_, k_] := T[n, k] = b[n, n][[k+1]] + If[k<2, 0, T[n, k-1]]; Table [Table [T[n, k], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Mar 11 2015, after Alois P. Heinz *)

T(n,k) = Sum_{j=1..k} A066633(n,j).

A092288 Triangle read by rows: T(n,k) = count of parts k in all plane partitions of n.

Original entry on oeis.org

1, 4, 1, 11, 2, 1, 28, 7, 2, 1, 62, 15, 5, 2, 1, 137, 38, 13, 5, 2, 1, 278, 76, 28, 11, 5, 2, 1, 561, 164, 60, 26, 11, 5, 2, 1, 1080, 316, 124, 52, 24, 11, 5, 2, 1, 2051, 623, 244, 108, 50, 24, 11, 5, 2, 1, 3778, 1156, 469, 208, 100, 48, 24, 11, 5, 2, 1, 6885, 2160, 886, 404, 194, 98, 48, 24, 11, 5, 2, 1

Offset: 1

Wouter Meeussen, Feb 01 2004

For large n the rows end in A091360 = partial sums of A000219 (count of plane partitions).

			Triangle begins:
    1;
    4,  1;
   11,  2,  1;
   28,  7,  2,  1;
   62, 15,  5,  2,  1;
  137, 38, 13,  5,  2,  1;
  ...

Alois P. Heinz, Rows n = 1..50, flattened
M. F. Hasler, A092288, rows 1 - 18.

Column k=1 gives A090539.
Row sums give A319648.
T(2n+1,n+1) gives A091360.
Cf. A000219, A066633.

Table[Length /@ Split[Sort[Flatten[planepartitions[k]]]], {k, 12}]

A092288_row(n, c=vector(n), m, k)={for(i=1, #n=PlanePartitions(n), for(j=1,#m=n[i], for(i=1,#k=m[j], c[k[i]]++))); c} \\ See A091298 for PlanePartitions(). See below for more efficient code.
M92288=[]; A092288(n,k,L=0)={n>1||return(if(L,[n,n==k],n==k)); if(#L&& #L<3, my(j=setsearch(M92288,[[n,k,L],[]],1)); j<=#M92288&& M92288[j][1]==[n,k,L]&& return(M92288[j][2])); my(c(p)=sum(i=1,#p,p[i]==k),S=[0,0],t); for(m=1,n,my(P=if(L,select(p->vecmin(L-Vecrev(p,#L))>=0, partitions(m,L[1],#L)), partitions(m))); if(mA092288(n-m,k,Vecrev(P[i])); S+=[t[1], t[1]*c(P[i])+t[2]], S+=[#P,vecsum(apply(c,P))])); if(L, #L<3&& M92288= setunion(M92288,[[[n,k,L],S]]);S,S[2])} \\ M. F. Hasler, Sep 26 2018

A299768 Triangle read by rows: T(n,k) = sum of all squares of the parts k in all partitions of n, with n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 2, 4, 4, 4, 9, 7, 12, 9, 16, 12, 16, 18, 16, 25, 19, 32, 36, 32, 25, 36, 30, 44, 54, 48, 50, 36, 49, 45, 76, 81, 96, 75, 72, 49, 64, 67, 104, 135, 128, 125, 108, 98, 64, 81, 97, 164, 189, 208, 200, 180, 147, 128, 81, 100, 139, 224, 279, 288, 300, 252, 245, 192, 162, 100, 121

Offset: 1

Omar E. Pol, Mar 19 2018

			Triangle begins:
   1;
   2,  4;
   4,  4,  9;
   7, 12,  9, 16;
  12, 16, 18, 16, 25,
  19, 32, 36, 32, 25, 36;
  30, 44, 54, 48, 50, 36, 49;
...
For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1], so the squares of the parts are respectively [16], [4, 4], [9, 1], [4, 1, 1], [1, 1, 1, 1]. The sum of the squares of the parts 1 is 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7. The sum of the squares of the parts 2 is 4 + 4 + 4 = 12. The sum of the squares of the parts 3 is 9. The sum of the squares of the parts 4 is 16. So the fourth row of triangle is [7, 12, 9, 16].

Alois P. Heinz, Rows n = 1..200, flattened

Column 1 is A000070.
Leading diagonal is A000290, n >= 1.
Row sums give A066183.
Both A180681 and A206561 have the same row sums as this triangle.
Cf. A066633, A138785.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1+n*x, b(n, i-1)+
      (p-> p+(coeff(p, x, 0)*i^2)*x^i)(b(n-i, min(n-i, i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Mar 20 2018

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1 + n*x, b[n, i - 1] + # + (Coefficient[#, x, 0]*i^2*x^i)&[b[n - i, Min[n - i, i]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 1, n}]&[b[n, n]];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, May 22 2018, after Alois P. Heinz *)

row(n) = {v = vector(n); forpart(p=n, for(k=1, #p, v[p[k]] += p[k]^2;);); v;} \\ Michel Marcus, Mar 20 2018

T(n,k) = (k^2)*A066633(n,k) = k*A138785(n,k). - Omar E. Pol, Jun 07 2018

More terms from Michel Marcus, Mar 20 2018

A325501 Product of Heinz numbers over all integer partitions of n.

Original entry on oeis.org

1, 2, 12, 240, 120960, 638668800, 15064408719360000, 27259975545259032576000000, 682714624600511148826789083611136000000000, 2948964060660649503322235948384635104494106968064000000000000000

Offset: 0

Gus Wiseman, May 06 2019

nonn

Row-products of A215366 (positive integers arranged by sum of prime indices A056239).

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The integer partitions of 3 are {(3), (2,1), (1,1,1)}, with Heinz numbers {5,6,8}, with product 240, so a(3) = 240.
The sequence of terms together with their prime indices begins:
          1: {}
          2: {1}
         12: {1,1,2}
        240: {1,1,1,1,2,3}
     120960: {1,1,1,1,1,1,1,2,2,2,3,4}
  638668800: {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5}

Index entries for sequences related to Heinz numbers

Cf. A003963, A006128, A007870, A008284, A056239, A066186, A066633, A112798, A118914, A124010, A145519, A215366, A325500, A325506, A325507.

Table[Times@@Prime/@(Join@@IntegerPartitions[n]),{n,0,5}]

A001222(a(n)) = A006128(n).
A056239(a(n)) = A066186(n).
A003963(a(n)) = A007870(n).
A124010(a(n),i) = A066633(n,i).

A325507 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all integer partitions of n.

Original entry on oeis.org

1, 2, 6, 28, 340, 3108, 106932, 2732340, 236790060, 19703562780, 3419598096420, 674127752953380, 264134168649181380, 95825592671995399620, 67662122741507082338220, 50556978553034312461203420, 69259146896604886347745839660, 104191622563656655781003976625020

Offset: 0

Gus Wiseman, May 07 2019

nonn

Also the Heinz number of row n of A066633.

The Heinz number of an integer partition or sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The integer partitions of 4 are {(4), (3,1), (2,2), (2,1,1), (1,1,1,1)}, with multiset union {1,1,1,1,1,1,1,2,2,2,3,4}, with multiplicities (7,3,1,1), so a(4) = prime(7)*prime(3)*prime(1)*prime(1) = 340.
The sequence of terms together with their prime indices begins:
                        1: {}
                        2: {1}
                        6: {1,2}
                       28: {1,1,4}
                      340: {1,1,3,7}
                     3108: {1,1,2,4,12}
                   106932: {1,1,2,4,8,19}
                  2732340: {1,1,2,3,6,11,30}
                236790060: {1,1,2,3,6,9,19,45}
              19703562780: {1,1,2,3,5,8,15,26,67}
            3419598096420: {1,1,2,3,5,8,13,21,41,97}
          674127752953380: {1,1,2,3,5,7,12,18,31,56,139}
       264134168649181380: {1,1,2,3,5,7,12,17,28,45,83,195}
     95825592671995399620: {1,1,2,3,5,7,11,16,25,38,63,112,272}
  67662122741507082338220: {1,1,2,3,5,7,11,16,24,35,55,87,160,373}

Index entries for sequences related to Heinz numbers

Cf. A001222, A003963, A006128, A007870, A056239, A066633, A087009, A112798, A215366, A302246, A325500, A325501, A325513.

Table[Times@@Prime/@Length/@Split[Sort[Join@@IntegerPartitions[n]]],{n,0,15}]

a(n) = Product_{i = 1..n} prime(A066633(n,i)).
a(n) = A181819(A003963(A325500(n))).
a(n) = A181819(A325501(n)).
A001222(a(n)) = n.
A056239(a(n)) = A006128(n).
For n > 0, A181819(a(n)) = A087009(n + 1).

A340057 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the block m consists of the divisors of m multiplied by A000041(n-m), with 1 <= m <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 3, 3, 2, 4, 1, 3, 1, 2, 4, 5, 3, 6, 2, 6, 1, 2, 4, 1, 5, 7, 5, 10, 3, 9, 2, 4, 8, 1, 5, 1, 2, 3, 6, 11, 7, 14, 5, 15, 3, 6, 12, 2, 10, 1, 2, 3, 6, 1, 7, 15, 11, 22, 7, 21, 5, 10, 20, 3, 15, 2, 4, 6, 12, 1, 7, 1, 2, 4, 8, 22, 15, 30, 11, 33, 7, 14, 28, 5, 25

Offset: 1

Omar E. Pol, Dec 27 2020

This triangle is a condensed version of the more irregular triangle A340035.

For further information about the correspondence divisor/part see A338156.

			Triangle begins:
  [1];
  [1],  [1, 2];
  [2],  [1, 2],  [1, 3];
  [3],  [2, 4],  [1, 3],  [1, 2, 4];
  [5],  [3, 6],  [2, 6],  [1, 2, 4],  [1, 5];
  [7],  [5, 10], [3, 9],  [2, 4, 8],  [1, 5],  [1, 2, 3, 6];
  [11], [7, 14], [5, 15], [3, 6, 12], [2, 10], [1, 2, 3, 6], [1, 7];
  ...
Row sums gives A066186.
Written as a tetrahedrons the first five slices are:
  --
  1;
  --
  1,
  1, 2;
  -----
  2,
  1, 2,
  1, 3;
  -----
  3,
  2, 4,
  1, 3,
  1, 2, 4;
  --------
  5,
  3, 6,
  2, 6,
  1, 2, 4,
  1, 5;
  --------
Row sums give A221529.
The slices of the tetrahedron appear in the upper zone of the following table (formed by four zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
|   |    -    |     |       |         |           |  5          |
| C |    -    |     |       |         |  3        |  3 6        |
| O |    -    |     |       |  2      |  2 4      |  2   6      |
| N | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
| D | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
| D | A027750 |     |       |         |           |  1          |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A027750 |     |       |         |  1        |  1 2        |
| I | A027750 |     |       |         |  1        |  1 2        |
| S | A027750 |     |       |         |  1        |  1 2        |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A027750 |     |       |  1      |  1 2      |  1   3      |
| S | A027750 |     |       |  1      |  1 2      |  1   3      |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
| L | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
| I |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| N | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| K |         |  |  |  |\|  |  |\|\|  |  |\|\|\|  |  |\|\|\|\|  |
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
| A |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| R |         |     |       |  3      |  3 1      |  3 1 1      |
| T |         |     |       |         |  2 2      |  2 2 1      |
| I |         |     |       |         |  4        |  4 1        |
| T |         |     |       |         |           |  3 2        |
| I |         |     |       |         |           |  5          |
| O |         |     |       |         |           |             |
| N |         |     |       |         |           |             |
| S |         |     |       |         |           |             |
|---|---------|-----|-------|---------|-----------|-------------|
.
The upper zone is a condensed version of the "divisors" zone.
The above table is the table of A340056 upside down.

Paolo Xausa, Table of n, a(n) for n = 1..11528 (rows 1..75 of the triangle, flattened)

Nonzero terms of A221649.

Cf. A000041, A002260, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A221529, A221530, A221531, A236104, A237593, A245092, A245095, A221650, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340011, A340031, A340032, A340056, A340057, A340061.

A340057row[n_]:=Flatten[Table[Divisors[m]PartitionsP[n-m],{m,n}]];Array[A340057row,10] (* Paolo Xausa, Sep 02 2023 *)

cp's OEIS Frontend

A340011 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the j-th row of triangle A127093 but with every term multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

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A340032 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the row m of triangle A127093, with 1 <= m <= n.

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A340056 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the divisors of j multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

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A015716 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, one of which is k (1<=k<=n).

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A210947 Triangle read by rows: T(n,k) = total number of parts <= k of all partitions of n.

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A092288 Triangle read by rows: T(n,k) = count of parts k in all plane partitions of n.

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A299768 Triangle read by rows: T(n,k) = sum of all squares of the parts k in all partitions of n, with n >= 1, 1 <= k <= n.

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A325501 Product of Heinz numbers over all integer partitions of n.