A135010 -id:A135010 - cp's OEIS frontend

A138138 A shell model of partitions. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.

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

Offset: 1

Omar E. Pol, Mar 16 2008, Mar 25 2008

The Integrated Diagram of Partitions is a shell model of partitions of a number. Partitions of n contains all partitions of the previous numbers. The number of shells of the partitions of n is equal to n. The number of parts of the last section of the set of partitions of n is A138137(n)=A006128(n)-A006128(n-1) and equal to the number of terms of row n. The number of terms of row n that are equal to 1 is A000041(n-1). The last term of row n is n. The shell model of partitions has several 2D and 3D versions.

			........................................
.. Integrated Diagram of Partitions ...
........... for n = 1 to 9 ............
.......................................
Partition number \ n = 1 2 3 4 5 6 7 8 9
........................................
.1) A000041(1)= 1 .... 1 1 1 1 1 1 1 1 1
.2) A000041(2)= 2 .... . 2 1 1 1 1 1 1 1
.3) A000041(3)= 3 .... . . 3 1 1 1 1 1 1
.4) .................. . 2 . 2 1 1 1 1 1
.5) A000041(4)= 5 .... . . . 4 1 1 1 1 1
.6) .................. . . 3 . 2 1 1 1 1
.7) A000041(5)= 7 .... . . . . 5 1 1 1 1
.8) .................. . 2 . 2 . 2 1 1 1
.9) .................. . . 3 . . 3 1 1 1
10) .................. . . . 4 . 2 1 1 1
11) A000041(6)=11 .... . . . . . 6 1 1 1
12) .................. . . 3 . 2 . 2 1 1
13) .................. . . . 4 . . 3 1 1
14) .................. . . . . 5 . 2 1 1
15) A000041(7)=15 .... . . . . . . 7 1 1
16) .................. . 2 . 2 . 2 . 2 1
17) .................. . . 3 . . 3 . 2 1
18) .................. . . . 4 . 2 . 2 1
19) .................. . . . 4 . . . 4 1
20) .................. . . . . 5 . . 3 1
21) .................. . . . . . 6 . 2 1
22) A000041(8)=22 .... . . . . . . . 8 1
23) .................. . . 3 . 2 . 2 . 2
24) .................. . . 3 . . 3 . . 3
25) .................. . . . 4 . . 3 . 2
26) .................. . . . . 5 . 2 . 2
27) .................. . . . . 5 . . . 4
28) .................. . . . . . 6 . . 3
29) .................. . . . . . . 7 . 2
30) A000041(9)=30 .... . . . . . . . . 9
.......................................
Triangle begins:
1
1,2
1,1,3,
1,1,1,2,2,4
1,1,1,1,1,2,3,5
1,1,1,1,1,1,1,2,2,2,3,3,2,4,6
1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,2,5,7
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,2,2,4,4,4,3,5,2,6,8
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,2,3,4,2,2,5,4,5,3,6,2,7,9

Robert Price, Table of n, a(n) for n = 1..4630, 20 rows.

Cf. A000041, A006128, A138137. See A135010 for another version.

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

A168016 Triangle T(n,k) read by rows in which row n list the number of partitions of n into parts divisible by k for k=n,n-1,...,1.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 0, 2, 5, 1, 0, 0, 0, 7, 1, 0, 0, 2, 3, 11, 1, 0, 0, 0, 0, 0, 15, 1, 0, 0, 0, 2, 0, 5, 22, 1, 0, 0, 0, 0, 0, 3, 0, 30, 1, 0, 0, 0, 0, 2, 0, 0, 7, 42, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 1, 0, 0, 0, 0, 0, 2, 0, 3, 5, 11, 77, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101

Offset: 1

Omar E. Pol, Nov 21 2009

			Triangle begins:
==============================================
.... k: 12 11 10. 9. 8. 7. 6. 5. 4. 3.. 2.. 1.
==============================================
n=1 ....................................... 1,
n=2 ................................... 1,  2,
n=3 ............................... 1,  0,  3,
n=4 ............................ 1, 0,  2,  5,
n=5 ......................... 1, 0, 0,  0,  7,
n=6 ...................... 1, 0, 0, 2,  3, 11,
n=7 ................... 1, 0, 0, 0, 0,  0, 15,
n=8 ................ 1, 0, 0, 0, 2, 0,  5, 22,
n=9 ............. 1, 0, 0, 0, 0, 0, 3,  0, 30,
n=10 ......... 1, 0, 0, 0, 0, 2, 0, 0,  7, 42,
n=11 ...... 1, 0, 0, 0, 0, 0, 0, 0, 0,  0, 56,
n=12 ... 1, 0, 0, 0, 0, 0, 2, 0, 3, 5, 11, 77,
...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000005, A000007, A000041, A035363, A035444, A047968, A135010.

Cf. A138121, A168014, A168015, A168020, A168021, A168111.

T[n_, k_]:= If[IntegerQ[n/(n-k+1)], PartitionsP[n/(n-k+1)], 0];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)

def T(n,k): return number_of_partitions(n/(n-k+1)) if (n%(n-k+1))==0 else 0
flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jan 12 2023

T(n, k) = A000041(n/k) if k|n; otherwise T(n,k) = 0.
T(n, n) = A000041(n).
From G. C. Greubel, Jan 12 2023: (Start)
T(2*n, n) = A000007(n-1).
Sum_{k=1..n} T(n, k) = A047968(n).
Sum_{k=2..n-1} T(n, k) = A168111(n-1). (End)

Edited and extended by Max Alekseyev, May 07 2010

A168017 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n listed in decreasing order.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 5, 1, 7, 1, 2, 3, 11, 1, 15, 1, 2, 5, 22, 1, 3, 30, 1, 2, 7, 42, 1, 56, 1, 2, 3, 5, 11, 77, 1, 101, 1, 2, 15, 135, 1, 3, 7, 176, 1, 2, 5, 22, 231, 1, 297, 1, 2, 3, 11, 30, 385, 1, 490, 1, 2, 5, 7, 42, 627, 1, 3, 15, 792, 1, 2, 56, 1002

Offset: 1

Omar E. Pol, Nov 22 2009

Positive values of triangle A168016.
The number of terms of row n is equal to the number of divisors of n: A000005(n).
Note that the last term of each row is the number of partitions of n: A000041(n).
Also, it appears that row n lists the partition numbers of the divisors of n. [Omar E. Pol, Nov 23 2009]

			Consider row n=8: (1, 2, 5, 22). The divisors of 8 listed in decreasing order are 8, 4, 2, 1 (see A056538). There is 1 partition of 8 into parts divisible by 8. Also, there are 2 partitions of 8 into parts divisible by 4: {(8), (4+4)}; 5 partitions of 8 into parts divisible by 2: {(8), (6+2), (4+4), (4+2+2), (2+2+2+2)}; and 22 partitions of 8 into parts divisible by 1, because A000041(8)=22. Then row 8 is formed by 1, 2, 5, 22.
Triangle begins:
1;
1,  2;
1,  3;
1,  2,  5;
1,  7;
1,  2,  3, 11;
1, 15;
1,  2,  5, 22;
1,  3, 30;
1,  2,  7, 42;
1, 56;
1,  2,  3,  5, 11, 77;

Alois P. Heinz, Rows n = 1..1400, flattened
Omar E. Pol, Illustration of the partitions of n, for n = 1 .. 9

Row sums give A047968.

Cf. A000005, A000041, A056538, A135010, A138121, A168016, A168018, A168019, A168020, A168021.

with(numtheory):
b:= proc(n, i, d) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n, i-d, d) +b(n-i, i, d)
      fi
    end:
T:= proc(n) local l;
      l:= sort([divisors(n)[]],`>`);
      seq(b(n, n, l[i]), i=1..nops(l))
    end:
seq(T(n), n=1..30); # Alois P. Heinz, Oct 21 2011

b[n_, i_, d_] := b[n, i, d] = Which[n<0, 0, n==0, 1, i<1, 0, True, b[n, i - d, d] + b[n-i, i, d]]; T[n_] := Module[{l = Divisors[n] // Reverse}, Table[b[n, n, l[[i]]], {i, 1, Length[l]}]]; Table[T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Dec 03 2015, after Alois P. Heinz *)

A182700 Triangle T(n,k) = n*A000041(n-k), 0<=k<=n, read by rows.

Original entry on oeis.org

0, 1, 1, 4, 2, 2, 9, 6, 3, 3, 20, 12, 8, 4, 4, 35, 25, 15, 10, 5, 5, 66, 42, 30, 18, 12, 6, 6, 105, 77, 49, 35, 21, 14, 7, 7, 176, 120, 88, 56, 40, 24, 16, 8, 8, 270, 198, 135, 99, 63, 45, 27, 18, 9, 9, 420, 300, 220, 150, 110, 70, 50, 30, 20, 10, 10, 616, 462, 330, 242, 165, 121, 77, 55, 33

Offset: 0

Omar E. Pol, Nov 27 2010

T(n,k) is the sum of the parts of all partitions of n that contain k as a part, assuming that all partitions of n have 0 as a part: Thus, column 0 gives the sum of the parts of all partitions of n.
By definition all entries in row n>0 are divisible by n.
Row sums are 0, 2, 8, 21, 48, 95, 180, 315, 536, 873, 1390, 2145,...
The partitions of n+k that contain k as a part can be obtained by adding k to every partition of n assuming that all partitions of n have 0 as a part.
For example, the partitions of 6+k that contain k as a part are
k + 6
k + 3 + 3
k + 4 + 2
k + 2 + 2 + 2
k + 5 + 1
k + 3 + 2 + 1
k + 4 + 1 + 1
k + 2 + 2 + 1 + 1
k + 3 + 1 + 1 + 1
k + 2 + 1 + 1 + 1 + 1
k + 1 + 1 + 1 + 1 + 1 + 1
The partition number A000041(n) is also the number of partitions of m*(n+k) into parts divisible by m and that contain m*k as a part, with k>=0, m>=1, n>=0 and assuming that all partitions of n have 0 as a part.

			For n=7 and k=4 there are 3 partitions of 7 that contain 4 as a part. These partitions are (4+3)=7, (4+2+1)=7 and (4+1+1+1)=7. The sum is 7+7+7 = 7*3 = 21. By other way, the partition number of 7-4 is A000041(3) = p(3)=3, then 7*3 = 21, so T(7,4) = 21.
Triangle begins with row n=0 and columns 0<=k<=n :
0,
1, 1,
4, 2, 2,
9, 6, 3, 3,
20,12,8, 4, 4,
35,25,15,10,5, 5,
66,42,30,18,12,6, 6

Robert Price, Table of n, a(n) for n = 0..5150 (First 100 rows)

Cf. A000041, A027293, A135010, A138121.
Two triangles that are essentially the same as this are A027293 and A140207. - N. J. A. Sloane, Nov 28 2010
Row sums give A182704.

A182700 := proc(n,k) n*combinat[numbpart](n-k) ; end proc:
seq(seq(A182700(n,k),k=0..n),n=0..15) ;

Table[n*PartitionsP[n-k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert Price, Jun 23 2020 *)

PARI
```
A182700(n,k) = n*numbpart(n-k)
```

T(n,0) = A066186(n).
T(n,k) = A182701(n,k), n>=1 and k>=1.
T(n,n) = n = min { T(n,k); 0<=k<=n }.

A182701 Triangle T(n,k) = n*A000041(n-k) read by rows, 1 <= k <= n. Sum of the parts of all partitions of n that contain k as a part.

Original entry on oeis.org

1, 2, 2, 6, 3, 3, 12, 8, 4, 4, 25, 15, 10, 5, 5, 42, 30, 18, 12, 6, 6, 77, 49, 35, 21, 14, 7, 7, 120, 88, 56, 40, 24, 16, 8, 8, 198, 135, 99, 63, 45, 27, 18, 9, 9, 300, 220, 150, 110, 70, 50, 30, 20, 10, 10, 462, 330, 242, 165, 121, 77, 55, 33, 22, 11, 11, 672, 504, 360, 264, 180, 132, 84, 60, 36, 24, 12, 12

Offset: 1

Omar E. Pol, Nov 27 2010

By definition, the entries in row n are divisible by n.
Row sums are 1, 4, 12, 28, 60, 114, ... = n*A000070(n).
Column 1 is A228816. - Omar E. Pol, Sep 25 2013

			Triangle begins:
    1;
    2,   2;
    6,   3,   3;
   12,   8,   4,   4;
   25,  15,  10,   5,   5;
   42,  30,  18,  12,   6,   6;
   77,  49,  35,  21,  14,   7,   7;
  120,  88,  56,  40,  24,  16,   8,   8;
  198, 135,  99,  63,  45,  27,  18,   9,   9;
  300, 220, 150, 110,  70,  50,  30,  20,  10,  10;

Robert Price, Table of n, a(n) for n = 1..5050 (First 100 rows)

Cf. A000041, A027293, A066186, A135010, A138121, A182700, A182702.

A182701 := proc(n,k) n*combinat[numbpart](n-k) ; end proc:
seq(seq(A182701(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Nov 28 2010

T[n_, k_] := n PartitionsP[n - k];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 19 2019 *)

T(n,k) = A182700(n,k), 1 <= k < n.

T(n,k) = n*A027293(n,k). - Omar E. Pol, Sep 25 2013

A182730 Even-indexed rows of triangle A141285.

Original entry on oeis.org

0, 2, 2, 4, 2, 4, 3, 6, 2, 4, 3, 6, 5, 4, 8, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12, 5, 4, 8, 7, 6, 11, 6, 5, 10, 9, 8, 7, 14, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12, 5, 4, 8, 7, 6, 11, 6, 5, 10, 9, 8, 7, 14, 4, 7, 6, 5, 10, 5, 9, 8, 7, 13, 4, 8, 7, 6, 12, 6, 11, 10, 9, 8, 16

Offset: 1

Omar E. Pol, Nov 28 2010, Nov 30 2010

			Triangle begins:
0,
2,
2, 4,
2, 4, 3, 6,
2, 4, 3, 6, 5, 4, 8,
2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10,
2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12

Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A135010, A138121, A141285, A182731, A182733.

Rows converge to A182732.

A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 10 2011

Row n lists the first A000041(n) terms of A141285.

The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic, see example. - Joerg Arndt, Sep 13 2013

			For n = 5 the partitions of 5 in colexicographic order are:
  1+1+1+1+1
  2+1+1+1
  3+1+1
  2+2+1
  4+1
  3+2
  5
so the fifth row is the largest in each partition: 1,2,3,2,4,3,5
Triangle begins:
  1;
  1,2;
  1,2,3;
  1,2,3,2,4;
  1,2,3,2,4,3,5;
  1,2,3,2,4,3,5,2,4,3,6;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7,2,4,3,6,5,4,8;
...

Wikiversity, Lexicographic and colexicographic order

The sum of row n is A006128(n).
Cf. A135010, A138121, A141285, A194547, A194548, A194549.
Row lengths are A000041.
Let y be the n-th integer partition in colexicographic order (A211992):
- The maximum of y is a(n).
- The length of y is A193173(n).
- The minimum of y is A196931(n).
- The Heinz number of y is A334437(n).
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Reverse-lexicographically ordered partitions are A080577.
Lexicographically ordered partitions are A193073.
Cf. A036037, A063008, A115623, A228100, A228531, A238966, A334301.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Max/@Join@@Table[Sort[IntegerPartitions[n],colex],{n,8}] (* Gus Wiseman, May 31 2020 *)

a(n) = A061395(A334437(n)). - Gus Wiseman, May 31 2020

Definition corrected by Omar E. Pol, Sep 12 2013

A207378 Triangle read by rows in which row n lists the parts of the last section of the set of partitions of n in nonincreasing order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 23 2012

Starting from the first row; it appears that the total numbers of occurrences of k in k successive rows give the sequence A000041. For more information see A182703.

			Written as a triangle:
1;
2,1;
3,1,1;
4,2,2,1,1,1;
5,3,2,1,1,1,1,1;
6,4,3,3,2,2,2,2,1,1,1,1,1,1,1;
7,5,4,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1;
8,6,5,4,4,4,3,3,3,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;

Alois P. Heinz, Rows n = 1..23, flattened

Triangle similar to A138121. Mirror of A207377. Row n has length A138137(n). Row sums give A138879. Column 1 is A000027.

Cf. A006128, A066186, A135010, A182703, A182712-A182714, A194812.

A210952 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the partitions of n but with the partitions aligned to the right margin.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 3, 7, 9, 1, 3, 7, 12, 12, 1, 3, 7, 14, 21, 20, 1, 3, 7, 14, 24, 31, 25, 1, 3, 7, 14, 26, 40, 47, 38, 1, 3, 7, 14, 26, 43, 61, 66, 49, 1, 3, 7, 14, 26, 45, 70, 92, 93, 69, 1, 3, 7, 14, 26, 45, 73, 106, 130, 124, 87, 1, 3, 7, 14

Offset: 1

Omar E. Pol, Apr 22 2012

			For n = 6 the illustration shows the partitions of 6 aligned to the right margin and below the sums of the columns:
.
.                      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,  3,  7, 14, 21, 20
.
So row 6 lists 1, 3, 7, 14, 21, 20.
Triangle begins:
1;
1, 3;
1, 3, 5;
1, 3, 7,  9;
1, 3, 7, 12, 12;
1, 3, 7, 14, 21, 20;
1, 3, 7, 14, 24, 31, 25;
1, 3, 7, 14, 26, 40, 47, 38;
1, 3, 7, 14, 26, 43, 61, 66, 49;
1, 3, 7, 14, 26, 45, 70, 92, 93, 69:

Mirror of triangle A206283. Rows sums give A066186. Rows converge to A014153. Right border gives A046746, >= 1.

Cf. A135010, A138121, A181187, A210763, A210950, A210951, A210953, A210961, A210970.

T(n,k) = Sum_{j=1..n} A210953(j,k). - Omar E. Pol, May 26 2012

A210990 Total area of the shadows of the three views of the shell model of partitions with n regions.

Original entry on oeis.org

0, 3, 10, 21, 26, 44, 51, 75, 80, 92, 99, 136, 143, 157, 166, 213, 218, 230, 237, 260, 271, 280, 348, 355, 369, 378, 403, 410, 427, 438, 526, 531, 543, 550, 573, 584, 593, 631, 640, 659, 672, 683, 804, 811, 825, 834, 859, 866, 883, 894, 938, 949, 958

Offset: 0

Omar E. Pol, Apr 23 2012

nonn

Each part is represented by a cuboid of sides 1 X 1 X k where k is the size of the part. For the definition of "regions of n" see A206437.

			For n = 11 the three views of the shell model of partitions with 11 regions look like this:
.
.   A182181(11) = 35            A182244(11) = 66
.
.   6                             * * * * * 6
.   3 3                      P    * * 3 * * 3
.   2   4                    a    * * * 4 * 2
.   2   2 2                  r    * 2 * 2 * 2
.   1       5                t    * * * * 5 1
.   1       2 3              i    * * 3 * 2 1
.   1       1   4            t    * * * 4 1 1
.   1       1   2 2          i    * 2 * 2 1 1
.   1       1   1   3        o    * * 3 1 1 1
.   1       1   1   1 2      n    * 2 1 1 1 1
.   1       1   1   1 1 1    s    1 1 1 1 1 1
. <------- Regions ------         ------------> N
.                            L
.                            a    1
.                            r    * 2
.                            g    * * 3
.                            e    * 2
.                            s    * * * 4
.                            t    * * 3
.                                 * * * * 5
.                            p    * 2
.                            a    * * * 4
.                            r    * * 3
.                            t    * * * * * 6
.                            s
.                               A182727(11) = 35
.
The areas of the shadows of the three views are A182244(11) = 66, A182181(11) = 35 and A182727(11) = 35, therefore the total area of the three shadows is 66+35+35 = 136, so a(11) = 136.

Other versions: A207380, A210970, A210979, A210980, A210991.

Cf. A135010, A138121, A141285, A194446, A182181, A182727, A186114, A206437.

a(n) = A182244(n) + A182727(n) + A182181(n), n >= 1.

a(A000041(n)) = 2*A006128(n) + A066186(n).

cp's OEIS Frontend

A138138 A shell model of partitions. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.

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A168016 Triangle T(n,k) read by rows in which row n list the number of partitions of n into parts divisible by k for k=n,n-1,...,1.

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A168017 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n listed in decreasing order.

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A182700 Triangle T(n,k) = n*A000041(n-k), 0<=k<=n, read by rows.

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A182701 Triangle T(n,k) = n*A000041(n-k) read by rows, 1 <= k <= n. Sum of the parts of all partitions of n that contain k as a part.

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A182730 Even-indexed rows of triangle A141285.

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A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

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A207378 Triangle read by rows in which row n lists the parts of the last section of the set of partitions of n in nonincreasing order.

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