This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A221529 #168 Aug 13 2025 23:03:16
%S A221529 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,
%T A221529 33,28,35,18,24,8,15,22,45,44,49,30,36,16,15,13,30,66,60,77,42,60,24,
%U A221529 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
%N A221529 Triangle read by rows: T(n,k) = A000203(k)*A000041(n-k), 1 <= k <= n.
%C A221529 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
%C A221529 From _Omar E. Pol_, Jul 10 2021: (Start)
%C A221529 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(h-1), for h = 1 to n, starting from the base of the tower, where n is the length of the largest side of the base.
%C A221529 The base of the tower is the symmetric representation of A024916(n).
%C A221529 The height of the tower is equal to A000041(n-1).
%C A221529 The surface area of the tower is equal to A345023(n).
%C A221529 The volume (or the number of cubes) of the tower equals A066186(n).
%C A221529 The volume represents the n-th term of the convolution of A000203 and A000041, that is A066186(n).
%C A221529 Note that the terraces that are the symmetric representation of sigma(n) and the terraces that are the symmetric representation of sigma(n-1) both are unified in level 1 of the structure. That is because the first two partition numbers A000041 are [1, 1].
%C A221529 The tower is an object of the family of the stepped pyramid described in A245092.
%C A221529 T(n,k) can be represented with a set of A237271(k) right prisms of height A000041(n-k) 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.
%C A221529 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(n-1) terms of the sequence A336811. Hence T(n,k) is also the sum of all divisors of all k's in the n-th row of triangle A176206.
%C A221529 The mentioned property is due to the correspondence between divisors and parts explained in A338156: all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.
%C A221529 Therefore the set of all partitions of n >= 1 has an associated tower.
%C A221529 The partial column sums of A340583 give this triangle showing the growth of the structure of the tower.
%C A221529 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)
%H A221529 Paolo Xausa, <a href="/A221529/b221529.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of triangle, flattened)
%H A221529 T. J. Osler, A. Hassen and T. R. Chandrupatia, <a href="https://citeseerx.ist.psu.edu/pdf/dfffc5ff66635132eee8d9d3cc77e50e1689ec45">Surprising connections between partitions and divisors</a>, The College Mathematics Journal, Vol. 38. No. 4, Sep. 2007, 278-287 (see p. 287).
%H A221529 Omar E. Pol, <a href="/A221529/a221529.jpg">Illustration of the prism, the tower and the 10th row of the triangle</a>
%F A221529 T(n,k) = sigma(k)*p(n-k) = A000203(k)*A027293(n,k).
%F A221529 T(n,k) = A245093(n,k)*A027293(n,k).
%e A221529 Triangle begins:
%e A221529 ------------------------------------------------------
%e A221529 n| k 1 2 3 4 5 6 7 8 9 10
%e A221529 ------------------------------------------------------
%e A221529 1| 1;
%e A221529 2| 1, 3;
%e A221529 3| 2, 3, 4;
%e A221529 4| 3, 6, 4, 7;
%e A221529 5| 5, 9, 8, 7, 6;
%e A221529 6| 7, 15, 12, 14, 6, 12;
%e A221529 7| 11, 21, 20, 21, 12, 12, 8;
%e A221529 8| 15, 33, 28, 35, 18, 24, 8, 15;
%e A221529 9| 22, 45, 44, 49, 30, 36, 16, 15, 13;
%e A221529 10| 30, 66, 60, 77, 42, 60, 24, 30, 13, 18;
%e A221529 ...
%e A221529 The sum of row 10 is [30 + 66 + 60 + 77 + 42 + 60 + 24 + 30 + 13 + 18] = A066186(10) = 420.
%e A221529 .
%e A221529 For n = 10 the calculation of the row 10 is as follows:
%e A221529 k A000203 T(10,k)
%e A221529 1 1 * 30 = 30
%e A221529 2 3 * 22 = 66
%e A221529 3 4 * 15 = 60
%e A221529 4 7 * 11 = 77
%e A221529 5 6 * 7 = 42
%e A221529 6 12 * 5 = 60
%e A221529 7 8 * 3 = 24
%e A221529 8 15 * 2 = 30
%e A221529 9 13 * 1 = 13
%e A221529 10 18 * 1 = 18
%e A221529 A000041
%e A221529 .
%e A221529 From _Omar E. Pol_, Jul 13 2021: (Start)
%e A221529 For n = 10 we can see below three views of two associated polycubes called here "prism of partitions" and "tower". Both objects contain the same number of cubes (that property is valid for n >= 1).
%e A221529 _ _ _ _ _ _ _ _ _ _
%e A221529 42 |_ _ _ _ _ |
%e A221529 |_ _ _ _ _|_ |
%e A221529 |_ _ _ _ _ _|_ |
%e A221529 |_ _ _ _ | |
%e A221529 |_ _ _ _|_ _ _|_ |
%e A221529 |_ _ _ _ | |
%e A221529 |_ _ _ _|_ | |
%e A221529 |_ _ _ _ _|_ | |
%e A221529 |_ _ _ | | |
%e A221529 |_ _ _|_ | | |
%e A221529 |_ _ | | | |
%e A221529 |_ _|_ _|_ _|_ _|_ | _
%e A221529 30 |_ _ _ _ _ | | | | 30
%e A221529 |_ _ _ _ _|_ | | | |
%e A221529 |_ _ _ | | | | |
%e A221529 |_ _ _|_ _ _|_ | | | |
%e A221529 |_ _ _ _ | | | | |
%e A221529 |_ _ _ _|_ | | | | |
%e A221529 |_ _ _ | | | | | |
%e A221529 |_ _ _|_ _|_ _|_ | | _|_|
%e A221529 22 |_ _ _ _ | | | | | 22
%e A221529 |_ _ _ _|_ | | | | |
%e A221529 |_ _ _ _ _|_ | | | | |
%e A221529 |_ _ _ | | | | | |
%e A221529 |_ _ _|_ | | | | | |
%e A221529 |_ _ | | | | | | |
%e A221529 |_ _|_ _|_ _|_ | | | _|_ _|
%e A221529 15 |_ _ _ _ | | | | | | | 15
%e A221529 |_ _ _ _|_ | | | | | | |
%e A221529 |_ _ _ | | | | | | | |
%e A221529 |_ _ _|_ _|_ | | | | _|_|_ _|
%e A221529 11 |_ _ _ | | | | | | | | 11
%e A221529 |_ _ _|_ | | | | | | | |
%e A221529 |_ _ | | | | | | | | |
%e A221529 |_ _|_ _|_ | | | | | _| |_ _ _|
%e A221529 7 |_ _ _ | | | | | | | | | 7
%e A221529 |_ _ _|_ | | | | | | _|_ _|_ _ _|
%e A221529 5 |_ _ | | | | | | | | | | | 5
%e A221529 |_ _|_ | | | | | | | _| | |_ _ _ _|
%e A221529 3 |_ _ | | | | | | | | _|_ _|_|_ _ _ _| 3
%e A221529 2 |_ | | | | | | | | | _ _|_ _|_|_ _ _ _ _| 2
%e A221529 1 |_|_|_|_|_|_|_|_|_|_| |_ _|_|_|_ _ _ _ _ _| 1
%e A221529 .
%e A221529 Figure 1. Figure 2.
%e A221529 Front view of the Lateral view
%e A221529 prism of partitions. of the tower.
%e A221529 .
%e A221529 . _ _ _ _ _ _ _ _ _ _
%e A221529 | | | | | | | | |_| 1
%e A221529 | | | | | | |_|_ _| 2
%e A221529 | | | | |_|_ |_ _| 3
%e A221529 | | |_|_ |_ _ _| 4
%e A221529 | |_ _ |_ |_ _ _| 5
%e A221529 |_ _ |_ |_ _ _ _| 6
%e A221529 |_ | |_ _ _ _| 7
%e A221529 |_ |_ _ _ _ _| 8
%e A221529 | | 9
%e A221529 |_ _ _ _ _ _| 10
%e A221529 .
%e A221529 Figure 3.
%e A221529 Top view
%e A221529 of the tower.
%e A221529 .
%e A221529 Figure 1 is a two-dimensional 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 that appears in the figures 2 and 3.
%e A221529 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(10-1) = 97.
%e A221529 The connection between these two associated objects is a representation of the correspondence divisor/part described in A338156. See also A336812.
%e A221529 The sum of the volumes of both objects equals A220909.
%e A221529 For the connection with the table of A338156 see also A340035. (End)
%t A221529 nrows=12; Table[Table[DivisorSigma[1,k]PartitionsP[n-k],{k,n}],{n,nrows}] // Flatten (* _Paolo Xausa_, Jun 17 2022 *)
%o A221529 (PARI) T(n,k)=sigma(k)*numbpart(n-k) \\ _Charles R Greathouse IV_, Feb 19 2013
%Y A221529 Row sums give A066186.
%Y A221529 Column 1 is A000041.
%Y A221529 Leading diagonals 1-5: A000203, A000203, A074400, A272027, A274535.
%Y A221529 Companion of A221530.
%Y A221529 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.
%K A221529 nonn,tabl,look
%O A221529 1,3
%A A221529 _Omar E. Pol_, Jan 20 2013