

A221529


Triangle read by rows: T(n,k) = A000203(k)*A000041(nk), 1 <= k <= n.


41



1, 1, 3, 2, 3, 4, 3, 6, 4, 7, 5, 9, 8, 7, 6, 7, 15, 12, 14, 6, 12, 11, 21, 20, 21, 12, 12, 8, 15, 33, 28, 35, 18, 24, 8, 15, 22, 45, 44, 49, 30, 36, 16, 15, 13, 30, 66, 60, 77, 42, 60, 24, 30, 13, 18, 42, 90, 88, 105, 66, 84, 40, 45, 26, 18, 12, 56, 126, 120, 154, 90, 132, 56, 75, 39, 36, 12, 28
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OFFSET

1,3


COMMENTS

Since A000203(k) has a symmetric representation, both T(n,k) and the partial sums of row n can be represented by symmetric polycubes. For more information see A237593 and A237270. For another version see A245099.  Omar E. Pol, Jul 15 2014
From Omar E. Pol, Jul 10 2021: (Start)
The above comment refers to a symmetric tower whose terraces are the symmetric representation of sigma(i), for i = 1..n, starting from the top. The levels of these terraces are the partition numbers A000041(h1), for h = 1 to n, starting from the base of the tower, where n is the length of the largest side of the base.
The base of the tower is the symmetric representation of A024916(n).
The height of the tower is equal to A000041(n1).
The surface area of the tower is equal to A345023(n).
The volume (or the number of cubes) of the tower equals A066186(n).
The volume represents the nth term of the convolution of A000203 and A000041, that is A066186(n).
Note that the terraces that are the symmetric representation of sigma(n) and the terraces that are the symmetric representation of sigma(n1) both are unified in level 1 of the structure. That is because the first two partition numbers A000041 are [1, 1].
The tower is an object of the family of the stepped pyramid described in A245092.
T(n,k) can be represented with a set of A237271(k) right prisms of height A000041(nk) since T(n,k) is the total number of cubes that are exactly below the parts of the symmetric representation of sigma(k) in the tower.
The tower is called here "tower of divisors" due to the following property:
T(n,k) is also the sum of all divisors of all k's that are in the first n rows of triangle A336811, or in other words, in the first A000070(n1) terms of the sequence A336811. Hence T(n,k) is also the sum of all divisors of all k's in the nth row of triangle A176206.
The mentioned property is due to the correspondence between divisors and parts explained in A338156: all divisors of the first A000070(n1) terms of A336811 are also all parts of all partitions of n.
Therefore the set of all partitions of n >= 1 has an associated tower of divisors.
The partial column sums of A340583 give this triangle showing the growth of the structure of the tower.
Note that the convolution of A000203 with any integer sequence S can be represented with a symmetric tower or structure of the same family where its terraces are the symmetric representation of sigma starting from the top and the heights of the terraces starting from the base are the terms of the sequence S. (End)


LINKS

Table of n, a(n) for n=1..78.
T. J. Osler, A. Hassen and T. R. Chandrupatia, Surprising connections between partitions and divisors, The College Mathematics Journal, Vol. 38. No. 4, Sep. 2007, 278287 (see p. 287).


FORMULA

T(n,k) = sigma(k)*p(nk) = A000203(k)*A027293(n,k).
T(n,k) = A245093(n,k)*A027293(n,k).


EXAMPLE

Triangle begins:

n k 1 2 3 4 5 6 7 8 9 10

1 1;
2 1, 3;
3 2, 3, 4;
4 3, 6, 4, 7;
5 5, 9, 8, 7, 6;
6 7, 15, 12, 14, 6, 12;
7 11, 21, 20, 21, 12, 12, 8;
8 15, 33, 28, 35, 18, 24, 8, 15;
9 22, 45, 44, 49, 30, 36, 16, 15, 13;
10 30, 66, 60, 77, 42, 60, 24, 30, 13, 18;
…
The sum of row 10 is [30 + 66 + 60 + 77 + 42 + 60 + 24 + 30 + 13 + 18] = A066186(10) = 420.
.
For n = 10 the calculation of the row 10 is as follows:
k A000203 T(10,k)
1 1 * 30 = 30
2 3 * 22 = 66
3 4 * 15 = 60
4 7 * 11 = 77
5 6 * 7 = 42
6 12 * 5 = 60
7 8 * 3 = 24
8 15 * 2 = 30
9 13 * 1 = 13
10 18 * 1 = 18
A000041
.
From Omar E. Pol, Jul 13 2021: (Start)
For n = 10 we can see below three views of two associated polycubes called here "prism of partitions" and "tower of divisors". Both objects contain the same number of cubes (that property is valid for n >= 1).
_ _ _ _ _ _ _ _ _ _
42 _ _ _ _ _ 
_ _ _ _ __ 
_ _ _ _ _ __ 
_ _ _ _  
_ _ _ __ _ __ 
_ _ _ _  
_ _ _ __  
_ _ _ _ __  
_ _ _   
_ _ __   
_ _    
_ __ __ __ __  _
30 _ _ _ _ _     30
_ _ _ _ __    
_ _ _     
_ _ __ _ __    
_ _ _ _     
_ _ _ __     
_ _ _      
_ _ __ __ __   __
22 _ _ _ _      22
_ _ _ __     
_ _ _ _ __     
_ _ _      
_ _ __      
_ _       
_ __ __ __    __ _
15 _ _ _ _        15
_ _ _ __       
_ _ _        
_ _ __ __     ___ _
11 _ _ _         11
_ _ __        
_ _         
_ __ __      _ _ _ _
7 _ _ _          7
_ _ __       __ __ _ _
5 _ _            5
_ __        _  _ _ _ _
3 _ _         __ ___ _ _ _ 3
2 _          _ __ ___ _ _ _ _ 2
1 __________ _ ____ _ _ _ _ _ 1
.
Figure 1. Figure 2.
Front view of the Lateral view of the
prism of partitions. tower of divisors.
.
. _ _ _ _ _ _ _ _ _ _
        _ 1
      __ _ 2
    __ _ _ 3
  __ _ _ _ 4
 _ _ _ _ _ _ 5
_ _ _ _ _ _ _ 6
_  _ _ _ _ 7
_ _ _ _ _ _ 8
  9
_ _ _ _ _ _ 10
.
Figure 3.
Top view of the
tower of divisors.
.
Figure 1 is a twodimensional diagram of the partitions of 10 in colexicographic order (cf. A026792, A211992). The area of the diagram is 10*42 = A066186(10) = 420. Note that the diagram can be interpreted also as the front view of a right prism whose volume is 1*10*42 = 420 equaling the volume and the number of cubes of the tower of divisors that appears in the figures 2 and 3.
Note that the shape and the area of the lateral view of the tower are the same as the shape and the area where the 1's are located in the diagram of partitions. In this case the mentioned area equals A000070(101) = 97.
The connection between these two associated objects is a representation of the correspondence divisor/part described in A338156. See also A336812.
The sum of the volumes of both objects equals A220909.
For the connection with the table of A338156 see also A340035. (End)


PROG

(PARI) T(n, k)=sigma(k)*numbpart(nk) \\ Charles R Greathouse IV, Feb 19 2013


CROSSREFS

Row sums give A066186.
Column 1 is A000041.
Leading diagonals 15: A000203, A000203, A074400, A272027, A274535.
Companion of A221530.
Cf. A000070, A000203, A026792, A027293, A135010, A138137, A176206, A182703, A220909, A211992, A221649, A236104, A237270, A237271, A237593, A245092, A245093, A245095, A245099, A262626, A336811, A336812, A338156, A339278, A340035, A340583, A340584, A345023, A346741.
Sequence in context: A093407 A349351 A147658 * A105161 A275769 A094365
Adjacent sequences: A221526 A221527 A221528 * A221530 A221531 A221532


KEYWORD

nonn,tabl


AUTHOR

Omar E. Pol, Jan 20 2013


STATUS

approved



