A207034 -id:A207034 - cp's OEIS frontend

A207038 Partial sums of A207034.

0, 1, 3, 5, 8, 11, 15, 18, 22, 26, 31, 35, 40, 45, 51, 55, 60, 65, 71, 77, 83, 90, 95, 101, 107, 114, 120, 127, 134, 142, 147, 153, 159, 166, 173, 180, 188, 195, 203, 211, 219, 228, 234, 241, 248, 256, 263, 271, 279, 288, 296, 304, 313, 322, 331, 341

Offset: 1

Omar E. Pol, Feb 21 2012

			Written as a triangle:
0;
1;
3;
5,8;
11,15;
18,22,26,31;
35,40,45,51;
55,60,65,71,77,83,90;
95,101,107,114,120,127,134,142;
147,153,159,166,173,180,188,195,203,211,219,228;
234,241,248,256,263,271,279,288,296,304,313,322,331,341;

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

Row n has length A187219(n). Right border gives A196087.

Cf. A006128, A135010, A138121, A194548, A207035.

A196087 Sum of all parts minus the total numbers of parts of all partitions of n.

Original entry on oeis.org

0, 1, 3, 8, 15, 31, 51, 90, 142, 228, 341, 525, 757, 1110, 1572, 2233, 3084, 4286, 5812, 7910, 10580, 14145, 18659, 24626, 32099, 41814, 53976, 69559, 88932, 113557, 143967, 182241, 229353, 288078, 360029, 449158, 557757, 691369, 853628, 1051974

Offset: 1

Omar E. Pol, Nov 10 2011

nonn

Also sum of parts of all partitions of n except the largest parts of the partitions. - Omar E. Pol, Feb 16 2012
Equals column 1 of A161224. - Omar E. Pol, Feb 26 2012
Partial sums of A207035. - Omar E. Pol, Apr 22 2012

			For n = 4 the five partitions of 4 are: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. The sum of all parts is 4+3+1+2+2+2+1+1+1+1+1+1 = 20. The sum of all parts is also the product n*p(n) = 4*5 = 20, where p(n) = A000041(n) is the number of partitions of n. On the other hand the number of parts in all partitions of 4 is equal to 12, so a(4) = 20-12 = 8.

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

Cf. A000041, A006128, A066186, A207034.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +g[1]*(i-1)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 20 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n==0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]*(i-1)}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 22 2015, after Alois P. Heinz *)

a(n) = n*numbpart(n) - sum(m=1, n, numdiv(m)*numbpart(n-m)); \\ Michel Marcus, Oct 22 2015

a(n) = n*A000041(n) - A006128(n) = A066186(n) - A006128(n).
a(n) = A207038(A000041(n)). - Omar E. Pol, Apr 21 2012
a(n) ~ exp(Pi*sqrt(2*n/3))/(4*sqrt(3)) * (1 - (3 + 6*gamma + Pi^2/24 + 3*log(6*n/Pi^2))/(Pi*sqrt(6*n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 24 2016
G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k)^2 / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021
a(n) = Sum_{k=1..n-1} p(n+j,j), where p(n,j) is the number of partitions of j having exactly j parts. E.g., a(4) = p(5,1) + p(6,2) + p(7,3) = 1+3+4 = 8. - Gregory L. Simay, Aug 19 2022

A228354 Indices (k) of partitions in the list of compositions of j in colexicographic order, if 1<=k<=2^(j-1), j>=1.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 22, 24, 28, 32, 44, 48, 56, 64, 86, 88, 92, 96, 112, 120, 128, 172, 176, 184, 192, 220, 224, 240, 256, 342, 344, 348, 352, 368, 376, 384, 440, 448, 480, 496, 512, 684, 688, 696, 704, 732, 736, 752, 768, 880, 888, 896, 960, 992, 1024

Offset: 1

Omar E. Pol, Aug 20 2013

Also where records occur in A228720.
Also triangle read by rows in which row j lists the indices of the partitions of j into parts greater than the smallest part of the partitions of j-1, in the list of compositions of j in colexicographic order. See A228525 and A211992.
The total number of terms in the first j rows of triangle is A000041(j), j >= 1.
Row j has length A187219(j).
Right border gives A000079.

			For j = 5 consider the list of compositions of 5 in colexicographic order (see A228525). The indices of the partitions are 1, 2, 4, 6, 8, 12, 16 which are the first A000041(5) terms of this sequence, see below:
---------------------------------------------------------
.   Compositions                     Partitions
k      of 5                             of 5      n  a(n)
---------------------------------------------------------
1    1+1+1+1+1  * ............... *  1+1+1+1+1    1    1
2    2+1+1+1    * ............... *  2+1+1+1      2    2
3    1+2+1+1          ........... *  3+1+1        3    4
4    3+1+1      * .../ .......... *  2+2+1        4    6
5    1+1+2+1          / ......... *  4+1          5    8
6    2+2+1      * .../ /   ...... *  3+2          6   12
7    1+3+1            /   /   ... *  5            7   16
8    4+1        * .../   /   /
9    1+1+1+2            /   /
10   2+1+2             /   /
11   1+2+2            /   /
12   3+2        * .../   /
13   1+1+3              /
14   2+3               /
15   1+4              /
16   5          * .../
.
Written as an irregular triangle the sequence begins:
1;
2;
4;
6,8;
12,16;
22,24,28,32;
44,48,56,64;
86,88,92,96,112,120,128;
172,176,184,192,220,224,240,256;
342,344,348,352,368,376,384,440,448,480,496,512;
684,688,696,704,732,736,752,768,880,888,896,960,992,1024;
...

Cf. A000041, A187219, A211992, A228354, A228525, A228720.

a(n) = 1 + A194602(n-1).
A001511(a(n)) = A141285(n).
A000120(a(n)-1) = A207034(n).

A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

Original entry on oeis.org

0, 1, 2, 5, 7, 16, 20, 39, 52, 86, 113, 184, 232, 353, 462, 661, 851, 1202, 1526, 2098, 2670, 3565, 4514, 5967, 7473, 9715, 12162, 15583, 19373, 24625, 30410, 38274, 47112, 58725, 71951, 89129, 108599, 133612, 162259, 198346, 239825, 291718, 351269, 425102

Offset: 1

Omar E. Pol, Feb 20 2012

nonn

			For n = 7 the last section of the set of partitions of 7 looks like this:
.
.        (. . . . . . 7)
.        (. . . 4 . . 3)
.        (. . . . 5 . 2)
.        (. . 3 . 2 . 2)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.
The sum of all parts = 7+4+3+5+2+3+2+2+1*11 = 39, on the other hand the total number of parts is 1+2+2+3+1*11 = 19, so a(7) = 39 - 19 = 20. Note that the number of dots in the picture is also equal to a(7) = 6+5+5+4 = 20.

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

Row sums of triangle A207034. Partial sums give A196087.

Cf. A006128, A066186, A135010, A138121, A138135, A138137, A138879, A138880, A187219, A194548, A207038.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<2 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +g[1]*(i-1)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50);  # Alois P. Heinz, Feb 20 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n==0, {1, 0}, i<2, {0, 0}, i>n , b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]*(i-1)}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 13 2015, after Alois P. Heinz *)

a(n) = A138879(n) - A138137(n) = A138880(n) - A138135(n). - Omar E. Pol, Apr 21 2012

G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k)^2 / Product_{j>=2} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

More terms from Alois P. Heinz, Feb 20 2012

A230440 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of partitions of n that do not contain 1 as a part in colexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 18 2013

The n-th row of triangle lists the parts of the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010.

			Illustration of initial terms (row = 1..6). The table shows the six sections of the set of partitions of 6 in three ways. Note that before the dissection, the set of partitions was in colexicographic order, see A211992. More generally, in a master model, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.
---------------------------------------------------------
n  j     Diagram          Parts              Parts
---------------------------------------------------------
.         _
1  1     |_|              1;                 1;
.           _
2  1      _| |              1,                 1,
2  2     |_ _|              2;               2;
.             _
3  1         | |              1,                 1,
3  2      _ _| |              1,               1,
3  3     |_ _ _|              3;             3;
.               _
4  1           | |              1,                 1,
4  2           | |              1,               1,
4  3      _ _ _| |              1,             1,
4  4     |_ _|   |            2,2,           2,2,
4  5     |_ _ _ _|              4;           4;
.                 _
5  1             | |              1,                 1,
5  2             | |              1,               1,
5  3             | |              1,             1,
5  4             | |              1,             1,
5  5      _ _ _ _| |              1,           1,
5  6     |_ _ _|   |            3,2,         3,2,
5  7     |_ _ _ _ _|              5;         5;
.                   _
6  1               | |              1,                 1,
6  2               | |              1,               1,
6  3               | |              1,             1,
6  4               | |              1,             1,
6  5               | |              1,           1,
6  6               | |              1,           1,
6  7      _ _ _ _ _| |              1,         1,
6  8     |_ _|   |   |          2,2,2,       2,2,2,
6  9     |_ _ _ _|   |            4,2,       4,2,
6  10    |_ _ _|     |            3,3,       3,3,
6  11    |_ _ _ _ _ _|              6;       6;
...
Triangle begins:
[1];
[1],[2];
[1],[1],[3];
[1],[1],[1],[2,2],[4];
[1],[1],[1],[1],[1],[3,2],[5];
[1],[1],[1],[1],[1],[1],[1],[2,2,2],[4,2],[3,3],[6];
...

Positive terms of A228716.
Row n has length A138137(n).
Row sums give A138879.
Right border gives A000027.
Cf. A000041, A135010, A138121, A141285, A182703, A187219, A193870, A194446, A206437, A207031, A207034, A207383, A207379, A211009.

A210946 Triangle read by rows: T(n,k) = sum of parts in the k-th column of the mirror of the last section of the set of partitions of n with its parts aligned to the right margin.

Original entry on oeis.org

1, 3, 5, 9, 2, 12, 3, 20, 9, 2, 25, 11, 3, 38, 22, 9, 2, 49, 28, 14, 3, 69, 44, 26, 9, 2, 87, 55, 37, 14, 3, 123, 83, 62, 29, 9, 2, 152

Offset: 1

Omar E. Pol, Apr 21 2012

Row n lists the positive terms of the n-th row of triangle A210953 in decreasing order.

			For n = 7 the illustration shows two arrangements of the last section of the set of partitions of 7:
.
.       (7)        (7)
.     (4+3)        (3+4)
.     (5+2)        (2+5)
.   (3+2+2)        (2+2+3)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.                 ---------
.                  25,11,3
.
The left hand picture shows the last section of 7 with its parts aligned to the right margin. In the right hand picture (the mirror) we can see that the sum of all parts of the columns 1..3 are 25, 11, 3 therefore row 7 lists 25, 11, 3.
Written as a triangle begins:
1;
3;
5;
9,    2;
12,   3;
20,   9,  2;
25,  11,  3;
38,  22,  9,  2;
49,  28, 14,  3;
69,  44, 26,  9,  2;
87,  55, 37, 14,  3,
123, 83, 62, 29,  9,  2;

Column 1 is A046746, n >= 1. Row sums give A138879.

Cf. A135010, A138121, A182703, A194714, A196807, A206437, A207031, A207034, A207035, A210945, A210952, A210953.

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A207038 Partial sums of A207034.

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A196087 Sum of all parts minus the total numbers of parts of all partitions of n.

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A228354 Indices (k) of partitions in the list of compositions of j in colexicographic order, if 1<=k<=2^(j-1), j>=1.

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A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

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A230440 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of partitions of n that do not contain 1 as a part in colexicographic order.

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A210946 Triangle read by rows: T(n,k) = sum of parts in the k-th column of the mirror of the last section of the set of partitions of n with its parts aligned to the right margin.

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