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

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, 8, 21, 24, 35, 24, 36, 16, 15, 9, 24, 28, 42, 30, 48, 24, 30, 13, 10, 27, 32, 49, 36, 60, 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

Offset

1,2

Comments

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.

In this case the sequence S is A000203.

The n-th diagonal of this triangle gives n times A000203.

The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.

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.

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.

The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.

For an illustration of the pyramid, see the Links section.

The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.

Column k lists the positive multiples of sigma(k).

The k-th term in the n-th diagonal is equal to n*sigma(k).

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.

Links

Indranil Ghosh, Rows 1..100 of triangle, flattened

Omar E. Pol, Illustration of the stepped pyramid with 16 levels

Formula

T(n,k) = (n-k+1) * A000203(k).

T(n,k) = A004736(n,k) * A245093(n,k).

Example

Triangle begins:
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;
8,  21, 24, 35, 24, 36,  16, 15;
9,  24, 28, 42, 30, 48,  24, 30,  13;
10, 27, 32, 49, 36, 60,  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;
14, 39, 48, 77, 60, 108, 64, 105, 78,  90,  48, 84,  28, 24;
15, 42, 52, 84, 66, 120, 72, 120, 91,  108, 60, 112, 42, 48, 24;
16, 45, 56, 91, 72, 132, 80, 135, 104, 126, 72, 140, 56, 72, 48, 31;
...
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.
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.
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).

Crossrefs

Row sums of triangle give A175254.

Diagonals 1-6: A000203, A074400, A272027, A239050, A274535, A274536.

Column 1 is A000027.

Initial zeros should be omitted in the following sequences related to the columns of triangle:

Columns 2-5: A008585, A008586, A008589, A008588.

Columns 6 and 11: A008594.

Columns 7-9: A008590, A008597, A008595.

Columns 10 and 17: A008600.

Columns 12-13: A135628, A008596.

Columns 14, 15 and 23: A008606.

Columns 16 and 25: A135631.

(There are many other OEIS sequences that are also columns of this triangle.)

Cf. A004736, A024916, A054973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A244050, A245092, A245093, A262612.

Sequence in context: A071915 A182578 A021432 * A141729 A272400 A238305

Adjacent sequences: A274821 A274822 A274823 * A274825 A274826 A274827

Keyword

nonn,tabl,easy

Author

Omar E. Pol, Oct 02 2016

Status

approved