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A340527 Triangle read by rows: T(n,k) = A024916(n-k+1)*A000041(k-1), 1 <= k <= n.

%I A340527 #34 Jun 01 2022 18:12:21
%S A340527 1,4,1,8,4,2,15,8,8,3,21,15,16,12,5,33,21,30,24,20,7,41,33,42,45,40,
%T A340527 28,11,56,41,66,63,75,56,44,15,69,56,82,99,105,105,88,60,22,87,69,112,
%U A340527 123,165,147,165,120,88,30,99,87,138,168,205,231,231,225,176,120,42,127,99,174
%N A340527 Triangle read by rows: T(n,k) = A024916(n-k+1)*A000041(k-1), 1 <= k <= n.
%C A340527 Conjecture 1: T(n,k) is the sum of divisors of the terms that are in the k-th blocks of the first n rows of triangle A176206.
%C A340527 Conjecture 2: the sum of row n equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.
%C A340527 Conjecture 3: T(n,k) is also the volume (or number of cubes) of the k-th block of a symmetric tower in which the terraces are the symmetric representation of sigma (n..1) starting from the base respectively (cf. A237270, A237593), hence the total area of the terraces is A024916(n), the same as the area of the base.
%C A340527 The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1). Hence the differences between levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).
%C A340527 This symmetric tower has the property that its volume (or total number of cubes) equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.
%C A340527 For another symmetric tower of the same family and whose volume equals A066186(n) see A339106 and A221529.
%C A340527 The above three conjectures are connected due to the correspondence between divisors and partitions (cf. A336811).
%e A340527 Triangle begins:
%e A340527    1;
%e A340527    4,   1;
%e A340527    8,   4,   2;
%e A340527   15,   8,   8,   3;
%e A340527   21,  15,  16,  12,   5;
%e A340527   33,  21,  30,  24,  20,   7;
%e A340527   41,  33,  42,  45,  40,  28,  11;
%e A340527   56,  41,  66,  63,  75,  56,  44,  15;
%e A340527   69,  56,  82,  99, 105, 105,  88,  60,  22;
%e A340527   87,  69, 112, 123, 165, 147, 165, 120,  88,  30;
%e A340527   99,  87, 138, 168, 205, 231, 231, 225, 176, 120,  42;
%e A340527 ...
%e A340527 For n = 6 the calculation of every term of row 6 is as follows:
%e A340527 --------------------------
%e A340527 k   A000041         T(6,k)
%e A340527 1      1  *  33   =   33
%e A340527 2      1  *  21   =   21
%e A340527 3      2  *  15   =   30
%e A340527 4      3  *   8   =   24
%e A340527 5      5  *   4   =   20
%e A340527 6      7  *   1   =    7
%e A340527 .          A024916
%e A340527 --------------------------
%e A340527 The sum of row 6 is 33 + 21 + 30 + 24 + 20 + 7 = 135, equaling A182738(6).
%Y A340527 Columns 1 and 2 give A024916.
%Y A340527 Column 3 gives A327329.
%Y A340527 Leading diagonal gives A000041.
%Y A340527 Row sums give A182738.
%Y A340527 Cf. A000070, A066186, A176206, A221529, A221531, A237270, A237593, A336811, A336812, A338156, A339106, A340035,  A340424, A340425, A340426, A340524, A340526.
%K A340527 nonn,tabl
%O A340527 1,2
%A A340527 _Omar E. Pol_, Jan 10 2021