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A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

%I A274824 #186 Dec 31 2020 11:11:15
%S A274824 1,2,3,3,6,4,4,9,8,7,5,12,12,14,6,6,15,16,21,12,12,7,18,20,28,18,24,8,
%T A274824 8,21,24,35,24,36,16,15,9,24,28,42,30,48,24,30,13,10,27,32,49,36,60,
%U A274824 32,45,26,18,11,30,36,56,42,72,40,60,39,36,12,12,33,40,63,48,84,48,75,52,54,24,28,13,36,44,70,54,96,56,90,65,72,36,56,14
%N A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.
%C A274824 Theorem: for any sequence S the partial sums of the partial sums are also the antidiagonal sums of the square array in which the n-th row gives n times the sequence S. Therefore they are also the row sums of the triangular array in which the n-th diagonal gives n times the sequence S.
%C A274824 In this case the sequence S is A000203.
%C A274824 The n-th diagonal of this triangle gives n times A000203.
%C A274824 The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.
%C A274824 T(n,k) is also the total number of unit cubes that are exactly below the terraces of the k-th level (starting from the top) up the base of the stepped pyramid with n levels described in A245092. This fact is because the mentioned terraces have the same shape as the symmetric representation of sigma(k). For more information see A237593 and A237270.
%C A274824 In the definition of this sequence the value n-k+1 is also the height of the terraces associated to sigma(k) in the mentioned pyramid with n levels, or in other words, the distance between the mentioned terraces and the base of the pyramid.
%C A274824 The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.
%C A274824 For an illustration of the pyramid, see the Links section.
%C A274824 The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.
%C A274824 Column k lists the positive multiples of sigma(k).
%C A274824 The k-th term in the n-th diagonal is equal to n*sigma(k).
%C A274824 Note that this is also a square array read by antidiagonals upwards: T(i,j) = i*sigma(j), i>=1, j>=1. The first row of the array is A000203. So consider that the pyramid is upside down. The value of "i" is the distance between the base of the pyramid and the terraces associated to sigma(j). The antidiagonal sums give the partial sums of the partial sums of A000203.
%H A274824 Indranil Ghosh, <a href="/A274824/b274824.txt"> Rows 1..100 of triangle, flattened</a>
%H A274824 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Illustration of the stepped pyramid with 16 levels</a>
%F A274824 T(n,k) = (n-k+1) * A000203(k).
%F A274824 T(n,k) = A004736(n,k) * A245093(n,k).
%e A274824 Triangle begins:
%e A274824 1;
%e A274824 2,  3;
%e A274824 3,  6,  4;
%e A274824 4,  9,  8,  7;
%e A274824 5,  12, 12, 14, 6;
%e A274824 6,  15, 16, 21, 12, 12;
%e A274824 7,  18, 20, 28, 18, 24,  8;
%e A274824 8,  21, 24, 35, 24, 36,  16, 15;
%e A274824 9,  24, 28, 42, 30, 48,  24, 30,  13;
%e A274824 10, 27, 32, 49, 36, 60,  32, 45,  26,  18;
%e A274824 11, 30, 36, 56, 42, 72,  40, 60,  39,  36,  12;
%e A274824 12, 33, 40, 63, 48, 84,  48, 75,  52,  54,  24, 28;
%e A274824 13, 36, 44, 70, 54, 96,  56, 90,  65,  72,  36, 56,  14;
%e A274824 14, 39, 48, 77, 60, 108, 64, 105, 78,  90,  48, 84,  28, 24;
%e A274824 15, 42, 52, 84, 66, 120, 72, 120, 91,  108, 60, 112, 42, 48, 24;
%e A274824 16, 45, 56, 91, 72, 132, 80, 135, 104, 126, 72, 140, 56, 72, 48, 31;
%e A274824 ...
%e A274824 For n = 16 and k = 10 the sum of the divisors of 10 is 1 + 2 + 5 + 10 = 18, and 16 - 10 + 1 = 7, and 7*18 = 126, so T(16,10) = 126.
%e A274824 On the other hand, the symmetric representation of sigma(10) has two parts of 9 cells, giving a total of 18 cells. In the stepped pyramid described in A245092, with 16 levels, there are 16 - 10 + 1 = 7 cubes exactly below every cell of the symmetric representation of sigma(10) up the base of pyramid; hence the total numbers of cubes exactly below the terraces of the 10th level (starting from the top) up the base of the pyramid is equal to 7*18 = 126. So T(16,10) = 126.
%e A274824 The sum of the 16th row of the triangle is 16 + 45 + 56 + 91 + 72 + 132 + 80 + 135 + 104 + 126 + 72 + 140 + 56 + 72 + 48 + 31 = A175254(16) = 1276, equaling the volume (also the number of cubes) of the stepped pyramid with 16 levels described in A245092 (see Links section).
%Y A274824 Row sums of triangle give A175254.
%Y A274824 Diagonals 1-6: A000203, A074400, A272027, A239050, A274535, A274536.
%Y A274824 Column 1 is A000027.
%Y A274824 Initial zeros should be omitted in the following sequences related to the columns of triangle:
%Y A274824 Columns 2-5: A008585, A008586, A008589, A008588.
%Y A274824 Columns 6 and 11: A008594.
%Y A274824 Columns 7-9: A008590, A008597, A008595.
%Y A274824 Columns 10 and 17: A008600.
%Y A274824 Columns 12-13: A135628, A008596.
%Y A274824 Columns 14, 15 and 23: A008606.
%Y A274824 Columns 16 and 25: A135631.
%Y A274824 (There are many other OEIS sequences that are also columns of this triangle.)
%Y A274824 Cf. A004736, A024916, A054973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A244050, A245092, A245093, A262612.
%K A274824 nonn,tabl,easy
%O A274824 1,2
%A A274824 _Omar E. Pol_, Oct 02 2016