A210970 -id:A210970 - cp's OEIS frontend

A210961 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which lists the sums of the columns of the shell model of partitions with n shells mentioned in A210970.

1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 5, 1, 1, 4, 3, 1, 1, 1, 2, 1, 5, 1, 1, 7, 3, 1, 1, 5, 3, 2, 1, 1, 1, 2, 1, 5, 1, 1, 9, 3, 1, 1, 12, 5, 2, 1, 1, 6, 6, 4, 2, 1, 1, 1, 2, 1, 5, 1, 1, 9, 3, 1, 1, 15, 5, 2, 1, 1, 15, 8, 4, 2, 1, 1, 7, 6, 5, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Apr 22 2012

			--------------------------------------------------------
Illustration of first five                      A210952
slices of the tetrahedron                       Row sum
--------------------------------------------------------
. 1,                                               1
.    1,                                            1
.    2, 1,                                         3
.          1,                                      1
.          2, 1,                                   3
.          3, 1, 1,                                5
.                   1,                             1
.                   2, 1,                          3
.                   5, 1, 1,                       7
.                   4, 3, 1, 1,                    9
.                               1,                 1
.                               2, 1,              3
.                               5, 1, 1,           7
.                               7, 3, 1, 1,       12
.                               5, 3, 2, 1, 1,    12
--------------------------------------------------------
. 1, 3, 1, 6, 2, 1,12, 5, 2, 1,20, 8, 4, 2, 1,
.
Column sums give A181187.
Also, written as a triangle read by rows in which each row is a flattened triangle, begins:
1;
1,2,1;
1,2,1,3,1,1;
1,2,1,5,1,1,4,3,1,1;
1,2,1,5,1,1,7,3,1,1,5,3,2,1,1;
1,2,1,5,1,1,9,3,1,1,12,5,2,1,1,6,6,4,2,1,1;
1,2,1,5,1,1,9,3,1,1,15,5,2,1,1,15,8,4,2,1,1,7,6,5,3,2,1,1;
In which row sums give A066186.

Cf. A066186, A135010, A138121, A181187, A210952, A210960, A210970.

A210960 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which list the number of parts in the columns of the shell model of partitions with n shells mentioned in A210970.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 3, 3, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 2, 1, 1, 3, 4, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Apr 22 2012

			--------------------------------------------------------
Illustration of first five
slices of the tetrahedron                       Row sum
--------------------------------------------------------
. 1,                                               1
.    1,                                            1
.    1, 1,                                         2
.          1,                                      1
.          1, 1,                                   2
.          1, 1, 1,                                3
.                   1,                             1
.                   1, 1,                          2
.                   2, 1, 1,                       4
.                   1, 2, 1, 1,                    5
.                               1,                 1
.                               1, 1,              2
.                               2, 1, 1,           4
.                               2, 2, 1, 1,        6
.                               1, 2, 2, 1, 1,     7
--------------------------------------------------------
. 1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 7, 6, 4, 2, 1,
.
It appears that column sums give A058399.
Also, written as a triangle read by rows in which each row is a flattened triangle, begins:
1;
1,1,1,
1,1,1,1,1,1;
1,1,1,2,1,1,1,2,1,1;
1,1,1,2,1,1,2,2,1,1,1,2,2,1,1;
1,1,1,2,1,1,3,2,1,1,3,3,2,1,1,1,3,3,2,1,1;
1,1,1,2,1,1,3,2,1,1,4,3,2,1,1,3,4,3,2,1,1,1,3,4,3,2,1,1;
In which row sums give A006128.

Cf. A058399, A135010, A138121, A182703, A209655, A209918, A210763, A210961.

A014153 Expansion of 1/((1-x)^2*Product_{k>=1} (1-x^k)).

Original entry on oeis.org

1, 3, 7, 14, 26, 45, 75, 120, 187, 284, 423, 618, 890, 1263, 1771, 2455, 3370, 4582, 6179, 8266, 10980, 14486, 18994, 24757, 32095, 41391, 53123, 67865, 86325, 109350, 137979, 173450, 217270, 271233, 337506, 418662, 517795, 638565, 785350, 963320, 1178628

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n with three kinds of 1. E.g., a(2)=7 because we have 2, 1+1, 1+1', 1+1", 1'+1', 1'+1", 1"+1". - Emeric Deutsch, Mar 22 2005
Partial sums of the partial sums of the partition numbers A000041. Partial sums of A000070. Euler transform of 3,1,1,1,...
Also sum of parts, counted without multiplicity, in all partitions of n, offset 1. Also Sum phi(p), where the sum is taken over all parts p of all partitions of n, offset 1. - Vladeta Jovovic, Mar 26 2005
Equals row sums of triangle A141157. - Gary W. Adamson, Jun 12 2008
A014153 convolved with A010815 = (1, 2, 3, ...). n-th partial sum sequence of A000041 convolved with A010815 = (n-1)-th column of Pascal's triangle, starting (1, n, ...). - Gary W. Adamson, Nov 09 2008
From Omar E. Pol, May 25 2012: (Start)
a(n) is also the sum of all parts of the (n+1)st column of a version of the section model of partitions in which every section has its parts aligned to the right margin (cf. A210953, A210970, A135010).
Rows of triangle A210952 converge to this sequence. (End)
Using the above result (see Jovovic's comment) of Jovovic and Mertens's theorem on the average order of the phi function, we can obtain the estimate a(n-1) = (6/Pi^2)*n*p(n) + O(log(n)*A006128(n)), where p(n) is the partition function A000041(n). It can be shown that A006128(n) = O(sqrt(n)*log(n)*p(n)), so we have the asymptotic result a(n) ~ (6/Pi^2)*n*p(n). - Peter Bala, Dec 23 2013
a(n-2) is the number of partitions of 2n or 2n-1 with palindromicity 2; that is, partitions that can be listed in palindromic order except for a central sequence of two distinct parts. - Gregory L. Simay, Nov 01 2015
Convolution of A000041 and A000027. - Omar E. Pol, Jun 17 2021
Convolution of A002865 and the positive terms of A000217. Partial sums give A014160. - Omar E. Pol, Mar 01 2023

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Mircea Merca and Maxie D. Schmidt, The partition function p(n) in terms of the classical Möbius function, arXiv:2310.13658 [math.CO], 2023.

Cf. A000041, A000070, A116930, A141157.
Cf. A010815. - Gary W. Adamson, Nov 09 2008
Column k=3 of A292508.
Cf. A000217, A002865, A014160.

m:=45; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/((1-x)^2*(&*[1-x^k: k in [1..50]])) )); // G. C. Greubel, Oct 15 2018

with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add((2+sigma(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 13 2012

a[n_] := a[n] = If[n == 0, 1, Sum[(2+DivisorSigma[1, j])*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
Table[Sum[(n-k)*PartitionsP[k],{k,0,n}],{n,1,50}] (* Vaclav Kotesovec, Jun 23 2015 *)
t[n_, k_] := Sum[StirlingS1[n, j]* Binomial[i + j - 1, i]* PartitionsP[k - n - i], {j, 0, n}, {i, 0, k - n}]; Print@ Table[t[n, k], {k, 10}, {n, 0, k - 1}]; Table[t[2, k], {k, 3, 43}] (* George Beck, May 25 2016 *)

x='x+O('x^45); Vec(1/((1-x)^2*prod(k=1,50, 1-x^k))) \\ G. C. Greubel, Oct 15 2018

Let t(n_, k_) = Sum_{i = 0..k} Sum_{j = 0..n} s(n, j)*C(i,  j)*p(k - n - i), where s(n, j) are Stirling numbers of the first kind, C(i, j) are the number of compositions of i distinct objects into j parts, and p is the integer partition function. Then a(k) = t(2, k+2) (conjectured). The formula for t(n, k) is the same as at A126442 except that there the Stirling numbers are of the second kind. - George Beck, May 21 2016
a(n) = (n+1)*A000070(n+1) - A182738(n+1). - Vaclav Kotesovec, Nov 04 2016
a(n) ~ exp(sqrt(2*n/3)*Pi)*sqrt(3)/(2*Pi^2) * (1 + 23*Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Nov 04 2016

A207380 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

Original entry on oeis.org

0, 3, 10, 21, 42, 70, 122, 187, 298, 443, 667, 957, 1401, 1960, 2775, 3828, 5295, 7167, 9745, 12998, 17380, 22915, 30196, 39347, 51274, 66126, 85209, 108942, 139055, 176273, 223148, 280733, 352623, 440646, 549597, 682411, 845852, 1044084, 1286512, 1579582

Offset: 0

Omar E. Pol, Feb 17 2012

nonn

In this model each part of a partition can be represented by a cuboid of size 1 x 1 x L, where L is the size of the part. One of the views is a rectangle formed by ones whose area is n*A000041(n) = A066186(n). Each element of the first view is equal to the volume of a horizontal column parallel to the axis x. The second view is the n-th slice illustrated in A026792 which has A000041(n) levels and its area is A006128(n) equals the total number of parts of all partitions of n and equals the sum of largest parts of all partitions of n. Each zone contains a partition of n. Each element of the second view is equal to the volume of a horizontal column parallel to the axis y. The third view is a triangle because it is also the n-th slice of the tetrahedron of A209655. The area of triangle is A000217(n). Each element of the third view is equal to the volume of a vertical column parallel to the axis z. The sum of elements of each view is A066186(n) equals the area of the first view. For more information about the shell model of partitions see A135010 and A182703.

			For n = 5 the three views of the three-dimensional shell model of partitions with 5 shells look like this:
.
.   A066186(5) = 35     A006128(5) = 20
.
.         1 1 1 1 1     5
.         1 1 1 1 1     3 2
.         1 1 1 1 1     4 1
.         1 1 1 1 1     2 2 1
.         1 1 1 1 1     3 1 1
.         1 1 1 1 1     2 1 1 1
.         1 1 1 1 1     1 1 1 1 1
.
.
.         7 6 4 2 1
.           1 2 3 2
.             1 1 2
.               1 1
.                 1
.
.   A000217(5) = 15
.
The areas of the shadows of the three views are A066186(5) = 35, A006128(5) = 20 and A000217(5) = 15, therefore the total area of the three shadows is 35+20+15 = 70, so a(5) = 70.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Other versions: A210970, A210979, A210980, A210990, A210991.

Cf. A000041, A000217, A006128, A026792, A066186, A135010, A138121, A141285, A182703, A182715, A206437, A209655.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+g[1]]
      fi
    end:
a:= n-> n*b(n, n)[1] +b(n, n)[2] +n*(n+1)/2:
seq (a(n), n=0..50);  # Alois P. Heinz, Mar 22 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If [n == 0 || i == 1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; Join[f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]] ]]]; a[n_] := n*b[n, n][[1]] + b[n, n][[2]] + n*(n+1)/2; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 18 2015, after Alois P. Heinz *)

a(n) = n * A000041(n) + A000217(n) + A006128(n) = A066186(n) + A000217(n) + A006128(n).

More terms from Alois P. Heinz, Mar 22 2012

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).

A210979 Total area of the shadows of the three views of the version "Tree" of the shell model of partitions with n shells.

Original entry on oeis.org

0, 3, 8, 15, 27, 42, 69, 102, 155, 225, 327, 458, 652, 894, 1232, 1669, 2257, 2999, 3996, 5242, 6877, 8928, 11564, 14845, 19045, 24223, 30756, 38815, 48877, 61195, 76496, 95124, 118067, 145930, 179991, 221160, 271268, 331538, 404463, 491948, 597253

Offset: 0

Omar E. Pol, Apr 28 2012

nonn

The physical model shows each part of a partition as an object, for example; a cube of side 1 which is labeled with the size of the part. Note that on the branches of the tree each column contains parts of the same size, as a periodic structure. For the large version of this model see A210980.

			For n = 7 the three views of the shell model of partitions version "tree" with seven shells looks like this:
.
.         A194805(7) = 25        A006128(7) = 54
.
.                        7       7
.                      4         4 3
.                    5           5 2
.                  3             3 2 2
.        6       1               6 1
.          3     1               3 3 1
.            4   1               4 2 1
.              2 1               2 2 2 1
.                1   5           5 1 1
.                1 3             3 2 1 1
.            4   1               4 1 1 1
.              2 1               2 2 1 1 1
.                1 3             3 1 1 1 1
.              2 1               2 1 1 1 1 1
.                1               1 1 1 1 1 1 1
-------------------------------------------------
.
.        6 3 4 2 1 3 5 4 7
.          3 2 2 1 2 2 3
.              2 1 2
.                1
.                1
.                1
.                1
.
.         A194803(7) = 23
.
The areas of the shadows of the three views are A006128(7) = 54, A194803(7) = 23 and A194805(7) = 25, therefore the total area of the three shadows is 54+23+25 = 102, so a(7) = 102.

Other versions: A207380, A210970, A210980, A210990, A210991.

Cf. A006128, A135010, A138121, A182703, A194803, A194805.

a(n) = A006128(n) + A194803(n) + A194805(n).

A210980 Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

Original entry on oeis.org

0, 3, 10, 21, 42, 69, 123, 189, 304, 458, 693, 998, 1474, 2067, 2927, 4056, 5613, 7595, 10335, 13782, 18411, 24276, 31944, 41583, 54152, 69762, 89758, 114668, 146181, 185083, 234051, 294126, 368992, 460669, 573906, 711865, 881506, 1087023, 1338043

Offset: 0

Omar E. Pol, Apr 21 2012

nonn

Each part is represented by a cuboid 1 X 1 X L where L is the size of the part.

			For n = 7 the shadows of the three views of the shell model of partitions version "tree" with seven shells looks like this:
.                                        |  Partitions
.    A194805(7) = 25    A066186(7) = 105 |  of 7
.                                        |
.                   1    * * * * * * 1   |  7
.                 2      * * * 1 * * 2   |  4+3
.               2        * * * * 1 * 2   |  5+2
.             3          * * 1 * 2 * 3   |  3+2+2
.   1       2            * * * * * 1 2   |  6+1
.     2     3            * * 1 * * 2 3   |  3+3+1
.       2   3            * * * 1 * 2 3   |  4+2+1
.         3 4            * 1 * 2 * 3 4   |  2+2+2+1
.           3   1        * * * * 1 2 3   |  5+1+1
.           4 2          * * 1 * 2 3 4   |  3+2+1+1
.       1   4            * * * 1 2 3 4   |  4+1+1+1
.         2 5            * 1 * 2 3 4 5   |  2+2+1+1+1
.           5 1          * * 1 2 3 4 5   |  3+1+1+1+1
.         1 6            * 1 2 3 4 5 6   |  2+1+1+1+1+1
.           7            1 2 3 4 5 6 7   |  1+1+1+1+1+1+1
.   ----------------------------------   |
.                                        |
.   * * * * 1 * * * *                    |
.   * * * 1 2 * * * *                    |
.   * 1 * * 2 1 * * *                    |
.   * * 1 2 2 * * 1 *                    |
.   * * * * 2 2 1 * *                    |
.   1 2 2 3 2 * * * *                    |
.           2 3 2 2 1                    |
.                                        |
.    A194804(7) = 59                     |
.
Note that, as a variant, in this case each part is labeled with its position in the partition.
The areas of the shadows of the three views are A066186(7) = 105, A194804(7) = 59 and A194805(7) = 25, therefore the total area of the three shadows is 105+59+25 = 189, so a(7) = 189.

Other versions: A207380, A210970, A210979, A210990, A210991.

Cf. A000041, A066186, A135010, A138121, A141285, A182703, A194804, A194805, A206437.

a(n) = A066186(n) + A194804(n) + A194805(n), n >= 1.

A210991 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, 9, 18, 21, 35, 39, 58, 61, 67, 71, 99, 103, 110, 115, 152, 155, 161, 165, 175, 181, 186, 238, 242, 249, 254, 265, 269, 277, 283, 352, 355, 361, 365, 375, 381, 386, 401, 406, 415, 422, 428, 522, 526, 533, 538, 549, 553, 561, 567, 584, 590, 595, 606

Offset: 0

Omar E. Pol, Apr 30 2012

nonn

It appears that if n is a partition number A000041 then the rotated structure with n regions shows each row as a partition of k such that A000041(k) = n (see example).

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           A210692(11) = 29
.
.   1                                       1
.   1                                       1
.   1                                       1
.   1                                       1
.   1       1                             1 1
.   1       1                             1 1
.   1       1   1                       1 1 1
.   2       1   1                       1 1 2
.   2       1   1   1                 1 1 1 2
.   3   2   2   2   1 1             1 1 2 2 3
.   6 3 4 2 5 3 4 2 3 2 1         1 2 3 4 5 6
. <------- 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 A182181(11) = 35, A182727(11) = 35 and A210692(11) = 29, therefore the total area of the three shadows is 35+35+29 = 99, so a(11) = 99.
Since n = 11 is a partition number A000041 we can see that the rotated structure with 11 regions shows each row as a partition of 6 because A000041(6) = 11. See below:
.
.                      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
.

Other versions: A207380, A210970, A210979, A210980, A210990.

Cf. A000041, A026905, A135010, A138121, A141285, A182703, A194446, A182181, A182727, A186114, A206437, A210692.

a(n) = A182181(n) + A182727(n) + A210692(n).

a(A000041(n)) = 2*A006128(n) + A026905(n).

A210950 Triangle read by rows: T(n,k) = number of parts in 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, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 4, 6, 7, 1, 2, 4, 7, 10, 11, 1, 2, 4, 7, 11, 14, 15, 1, 2, 4, 7, 12, 17, 21, 22, 1, 2, 4, 7, 12, 18, 25, 29, 30, 1, 2, 4, 7, 12, 19, 28, 36, 41, 42, 1, 2, 4, 7, 12, 19, 29, 40, 50, 55, 56, 1, 2, 4, 7, 12, 19, 30, 43

Offset: 1

Omar E. Pol, Apr 22 2012

Index of the first partition of n that has k parts, when the partitions of n are listed in reverse lexicographic order, as in Mathematica's IntegerPartitions[n]. - Clark Kimberling, Oct 16 2023

			For n = 6 the partitions of 6 aligned to the right margin look like this:
.
.                      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
.
The number of parts in columns 1-6 are
.  1,  2,  4,  7, 10, 11, the same as the 6th row of triangle.
Triangle begins:
  1;
  1, 2;
  1, 2, 3;
  1, 2, 4, 5;
  1, 2, 4, 6, 7;
  1, 2, 4, 7, 10, 11;
  1, 2, 4, 7, 11, 14, 15;
  1, 2, 4, 7, 12, 17, 21, 22;
  1, 2, 4, 7, 12, 18, 25, 29, 30;
  1, 2, 4, 7, 12, 19, 28, 36, 41, 42;
  1, 2, 4, 7, 12, 19, 29, 40, 50, 55, 56;
  1, 2, 4, 7, 12, 19, 30, 43, 58, 70, 76, 77;

Mirror of A058399. Row sums give A006128. Right border gives A000041, n >= 1. Rows converge to A000070.

Cf. A135010, A194714, A210945, A210951, A210952, A210953, A210970.

m[n_, k_] := Length[IntegerPartitions[n][[k]]]; c[n_] := PartitionsP[n];
t[n_, h_] := Select[Range[c[n]], m[n, #] == h &, 1];
Column[Table[t[n, h], {n, 1, 20}, {h, 1, n}]]
 (* Clark Kimberling, Oct 16 2023 *)

T(n,k) = Sum_{j=1..n} A210951(j,k).

cp's OEIS Frontend

A210961 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which lists the sums of the columns of the shell model of partitions with n shells mentioned in A210970.

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A210960 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which list the number of parts in the columns of the shell model of partitions with n shells mentioned in A210970.

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A014153 Expansion of 1/((1-x)^2*Product_{k>=1} (1-x^k)).

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A207380 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

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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.

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

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A210979 Total area of the shadows of the three views of the version "Tree" of the shell model of partitions with n shells.

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A210980 Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

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A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

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A210950 Triangle read by rows: T(n,k) = number of parts in the k-th column of the partitions of n but with the partitions aligned to the right margin.

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