A135010 -id:A135010 - cp's OEIS frontend

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

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

Offset: 0

Omar E. Pol, Jul 28 2013

nonn

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

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

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 29 2020

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

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

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

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

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

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

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

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

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

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

A206563 Triangle read by rows: T(n,k) = number of odd/even parts >= k in all partitions of n, if k is odd/even.

Original entry on oeis.org

1, 2, 1, 5, 1, 1, 8, 4, 1, 1, 15, 5, 3, 1, 1, 24, 11, 5, 3, 1, 1, 39, 15, 9, 4, 3, 1, 1, 58, 28, 13, 9, 4, 3, 1, 1, 90, 38, 23, 12, 8, 4, 3, 1, 1, 130, 62, 33, 21, 12, 8, 4, 3, 1, 1, 190, 85, 51, 29, 20, 11, 8, 4, 3, 1, 1, 268, 131, 73, 48, 28, 20, 11, 8, 4, 3, 1, 1

Offset: 1

Omar E. Pol, Feb 15 2012

Let m and n be two positive integers such that m <= n. It appears that any set formed by m connected sections, or m disconnected sections, or a mixture of both, has the same properties described in the section example. (Cf. A135010, A207031, A207032, A212010). - Omar E. Pol, May 01 2012

			Calculation for n = 6. Write the partitions of 6 and below the sums of their 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
. ------------------------
.  35, 16,  8,  4,  2,  1  --> Row 6 of triangle A181187.
.   |  /|  /|  /|  /|  /|
.   | / | / | / | / | / |
.   |/  |/  |/  |/  |/  |
.  19,  8,  4,  2,  1,  1  --> Row 6 of triangle A066633.
.
More generally, it appears that the sum of column k is also the total number of parts >= k in all partitions of n. It appears that the first differences of the column sums together with 1 give the number of occurrences of k in all partitions of n.
On the other hand we can see that the partitions of 6 contain:
24  odd parts >= 1 (the odd parts).
11 even parts >= 2 (the even parts).
5   odd parts >= 3.
3  even parts >= 4.
2   odd parts >= 5.
1  even part  >= 6.
Then, using the values of the column sums, it appears that:
T(6,1) = 35 - 16 + 8 - 4 + 2 - 1 = 24
T(6,2) =      16 - 8 + 4 - 2 + 1 = 11
T(6,3) =           8 - 4 + 2 - 1 = 5
T(6,4) =               4 - 2 + 1 = 3
T(6,5) =                   2 - 1 = 1
T(6,6) =                       1 = 1
So the 6th row of our triangle gives 24, 11, 5, 3, 1, 1.
Finally, for all partitions of 6, we can write:
The number of  odd parts      is equal to T(6,1) = 24.
The number of even parts      is equal to T(6,2) = 11.
The number of  odd parts >= 3 is equal to T(6,3) = 5.
The number of even parts >= 4 is equal to T(6,4) = 3.
The number of  odd parts >= 5 is equal to T(6,5) = 1.
The number of even parts >= 6 is equal to T(6,6) = 1.
More generally, we can write the same properties for any positive integer.
Triangle begins:
1;
2,    1;
5,    1,  1;
8,    4,  1,  1;
15,   5,  3,  1,  1;
24,  11,  5,  3,  1,  1;
39,  15,  9,  4,  3,  1,  1;
58,  28, 13,  9,  4,  3,  1,  1;
90,  38, 23, 12,  8,  4,  3,  1,  1;
130, 62, 33, 21, 12,  8,  4,  3,  1,  1;

Columns 1-2 give A066897, A066898.

Cf. A006128, A066633, A181187, A182703, A207031, A207032.

It appears that T(n,k) = abs(Sum_{j=k..n} (-1)^j*A181187(n,j)).

It appears that A066633(n,k) = T(n,k) - T(n,k+2). - Omar E. Pol, Feb 26 2012

More terms from Alois P. Heinz, Feb 18 2012

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

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 07 2013

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

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

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

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

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

A221530 Triangle read by rows: T(n,k) = A000005(k)*A000041(n-k).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 4, 2, 3, 5, 6, 4, 3, 2, 7, 10, 6, 6, 2, 4, 11, 14, 10, 9, 4, 4, 2, 15, 22, 14, 15, 6, 8, 2, 4, 22, 30, 22, 21, 10, 12, 4, 4, 3, 30, 44, 30, 33, 14, 20, 6, 8, 3, 4, 42, 60, 44, 45, 22, 28, 10, 12, 6, 4, 2, 56, 84, 60, 66, 30, 44, 14, 20, 9, 8, 2, 6

Offset: 1

Omar E. Pol, Jan 19 2013

T(n,k) is the number of partitions of n that contain k as a part multiplied by the number of divisors of k.
It appears that T(n,k) is also the total number of appearances of k in the last k sections of the set of partitions of n multiplied by the number of divisors of k.
T(n,k) is also the number of partitions of k into equal parts multiplied by the number of ones in the j-th section of the set of partitions of n, where j = (n - k + 1).
For another version see A245095. - Omar E. Pol, Jul 15 2014

			For n = 6:
  -------------------------
  k   A000005        T(6,k)
  1      1  *  7   =    7
  2      2  *  5   =   10
  3      2  *  3   =    6
  4      3  *  2   =    6
  5      2  *  1   =    2
  6      4  *  1   =    4
  .         A000041
  -------------------------
So row 6 is [7, 10, 6, 6, 4, 2]. Note that the sum of row 6 is 7+10+6+6+2+4 = 35 equals A006128(6).
.
Triangle begins:
  1;
  1,   2;
  2,   2,  2;
  3,   4,  2,  3;
  5,   6,  4,  3,  2;
  7,  10,  6,  6,  2,  4;
  11, 14, 10,  9,  4,  4,  2;
  15, 22, 14, 15,  6,  8,  2,  4;
  22, 30, 22, 21, 10, 12,  4,  4,  3;
  30, 44, 30, 33, 14, 20,  6,  8,  3,  4;
  42, 60, 44, 45, 22, 28, 10, 12,  6,  4,  2;
  56, 84, 60, 66, 30, 44, 14, 20,  9,  8,  2,  6;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)

Similar to A221529.
Columns 1-2: A000041, A139582. Leading diagonals 1-3: A000005, A000005, A062011. Row sums give A006128.
Cf. A027293, A135010, A138137, A182703, A245095, A245099.

A221530row[n_]:=DivisorSigma[0,Range[n]]PartitionsP[n-Range[n]];Array[A221530row,10] (* Paolo Xausa, Sep 04 2023 *)

row(n) = vector(n, i, numdiv(i)*numbpart(n-i)); \\ Michel Marcus, Jul 18 2014

T(n,k) = d(k)*p(n-k) = A000005(k)*A027293(n,k).

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

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Jul 29 2013

nonn

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

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

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

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

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

A087787 a(n) = Sum_{k=0..n} (-1)^(n-k)*A000041(k).

Original entry on oeis.org

1, 0, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409

Offset: 0

Vladeta Jovovic, Oct 07 2003

nonn

Essentially first differences of A024786 (see the formula). Also, a(n) is the number of 2's in the last section of the set of partitions of n+2 (see A135010). - Omar E. Pol, Sep 10 2008
From Gus Wiseman, May 20 2024: (Start)
Also the number of integer partitions of n containing an even number of ones, ranked by A003159. The a(0) = 1 through a(8) = 15 partitions are:
  ()  .  (2)   (3)  (4)     (5)    (6)       (7)      (8)
         (11)       (22)    (32)   (33)      (43)     (44)
                    (211)   (311)  (42)      (52)     (53)
                    (1111)         (222)     (322)    (62)
                                   (411)     (511)    (332)
                                   (2211)    (3211)   (422)
                                   (21111)   (31111)  (611)
                                   (111111)           (2222)
                                                      (3311)
                                                      (4211)
                                                      (22211)
                                                      (41111)
                                                      (221111)
                                                      (2111111)
                                                      (11111111)
Also the number of integer partitions of n + 1 containing an odd number of ones, ranked by A036554.
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A024786, A135010, A138121, A141285.
The unsigned version is A000070, strict A036469.
For powers of 2 instead number of partitions we have A001045.
The strict or odd version is A025147 or A096765.
The ordered version (compositions instead of partitions) is A078008.
For powers of 2 instead of -1 we have A259401, cf. A259400.
A002865 counts partitions with no ones, column k=0 of A116598.
A072233 counts partitions by sum and length.
Cf. A000009, A003159, A026804, A027187, A027193, A036554, A058695, A101707.

Table[Sum[(-1)^(n-k)*PartitionsP[k], {k,0,n}], {n,0,50}] (* Vaclav Kotesovec, Aug 16 2015 *)
(* more efficient program *) sig = 1; su = 1; Flatten[{1, Table[sig = -sig; su = su + sig*PartitionsP[n]; Abs[su], {n, 1, 50}]}] (* Vaclav Kotesovec, Nov 06 2016 *)
Table[Length[Select[IntegerPartitions[n], EvenQ[Count[#,1]]&]],{n,0,30}] (* Gus Wiseman, May 20 2024 *)

from sympy import npartitions
def A087787(n): return sum(-npartitions(k) if n-k&1 else npartitions(k) for k in range(n+1)) # Chai Wah Wu, Oct 25 2023

G.f.: 1/(1+x)*1/Product_{k>0} (1-x^k).
a(n) = 1/n*Sum_{k=1..n} (sigma(k)+(-1)^k)*a(n-k).
a(n) = A024786(n+2)-A024786(n+1). - Omar E. Pol, Sep 10 2008
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) * (1 + (11*Pi/(24*sqrt(6)) - sqrt(3/2)/Pi)/sqrt(n) - (11/16 + (23*Pi^2)/6912)/n). - Vaclav Kotesovec, Nov 05 2016
a(n) = A000041(n) - a(n-1). - Jon Maiga, Aug 29 2019
Alternating partial sums of A000041. - Gus Wiseman, May 20 2024

A168021 Triangle T(n,k) read by rows in which row n lists the number of partitions of n into parts divisible by k.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 20 2009, Nov 21 2009

The row-reversed version is A168016.

Also see A168020.

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

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Omar E. Pol, Illustration of the shell model of partitions (2D and 3D)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000007, A000012, A000041, A035377, A035444, A047968 (row sums).

Cf. A135010, A138121, A168016, A168017, A168018, A168019, A168020.

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

def A168021(n,k): return number_of_partitions(n/k) if (n%k)==0 else 0
flatten([[A168021(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, else T(n,k)=0.
Sum_{k=1..n} T(n, k) = A047968(n).
From G. C. Greubel, Jan 12 2023: (Start)
T(2*n, n) = 2*A000012(n).
T(2*n-1, n+1) = A000007(n-2). (End)

Edited by Charles R Greathouse IV, Mar 23 2010

A194436 Triangle read by rows: T(n,k) = number of parts in the k-th region of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 5, 1, 2, 3, 1, 5, 1, 7, 1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 1, 15, 1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 1, 15, 1, 2, 1, 4, 1, 1, 22, 1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 1, 15

Offset: 1

Omar E. Pol, Nov 27 2011

			Triangle begins:
1;
1,2;
1,2,3;
1,2,3,1,5;
1,2,3,1,5,1,7;
1,2,3,1,5,1,7,1,2,1,11;
1,2,3,1,5,1,7,1,2,1,11,1,2,1,15;
1,2,3,1,5,1,7,1,2,1,11,1,2,1,15,1,2,1,4,1,1,22;
...
Row n has length A000041(n). Row sums give A006128, n >= 1. Right border gives A000041, n >= 1. Records in every row give A000041, n >= 1. Rows converge to A194446.

Cf. A000041, A006128, A135010, A138121, A186114, A186412, A193870, A194437, A194438, A194439, A194446, A194447.

A194439 Number of regions in the set of partitions of n that contain only one part.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297

Offset: 1

Omar E. Pol, Nov 28 2011

It appears that this is 1 together with A000041. - Omar E. Pol, Nov 29 2011

For the definition of "region" see A206437. See also A186114 and A193870.

			For n = 5 the seven regions of 5 in nondecreasing order are the sets of positive integers of the rows as shown below:
   1;
   1, 2;
   1, 1, 3;
   0, 0, 0, 2;
   1, 1, 1, 2, 4;
   0, 0, 0, 0, 0, 3;
   1, 1, 1, 1, 1, 2, 5;
   ...
There are three regions that contain only one positive part, so a(5) = 3.
Note that in every column of the triangle the positive integers are also the parts of one of the partitions of 5.

Omar E. Pol, Illustration of the seven regions of 5

Column 1 of A194438.

Cf. A000041, A002865, A027336, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194446, A194447, A206437.

It appears that a(n) = A000041(n-2), if n >= 2. - Omar E. Pol, Nov 29 2011

It appears that a(n) = A000041(n) - A027336(n), if n >= 2. - Omar E. Pol, Nov 30 2011

Definition clarified by Omar E. Pol, May 21 2021

cp's OEIS Frontend

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

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

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A206563 Triangle read by rows: T(n,k) = number of odd/even parts >= k in all partitions of n, if k is odd/even.

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

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A221530 Triangle read by rows: T(n,k) = A000005(k)*A000041(n-k).

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

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A087787 a(n) = Sum_{k=0..n} (-1)^(n-k)*A000041(k).

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A168021 Triangle T(n,k) read by rows in which row n lists the number of partitions of n into parts divisible by k.

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