A272026 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers A016945 interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

3, 9, 15, 3, 21, 0, 27, 9, 33, 0, 3, 39, 15, 0, 45, 0, 0, 51, 21, 9, 57, 0, 0, 3, 63, 27, 0, 0, 69, 0, 15, 0, 75, 33, 0, 0, 81, 0, 0, 9, 87, 39, 21, 0, 3, 93, 0, 0, 0, 0, 99, 45, 0, 0, 0, 105, 0, 27, 15, 0, 111, 51, 0, 0, 0, 117, 0, 0, 0, 9, 123, 57, 33, 0, 0, 3, 129, 0, 0, 21, 0, 0, 135, 63, 0, 0, 0, 0, 141, 0, 39, 0, 0, 0

Offset: 1

Author

Omar E. Pol, Apr 18 2016

Keywords

• nonn
• tabf

Comments

Alternating sum of row n equals 3 times sigma(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = 3*A000203(n) = A272027(n).

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

The number of positive terms in row n is A001227(n).

If T(n,k) = 9 then T(n+1,k+1) = 3 is the first element of the column k+1.

For more information see A196020.

Examples

			Triangle begins:
    3;
    9;
   15,  3;
   21,  0;
   27,  9;
   33,  0,  3;
   39, 15,  0;
   45,  0,  0;
   51, 21,  9;
   57,  0,  0,  3;
   63, 27,  0,  0;
   69,  0, 15,  0;
   75, 33,  0,  0;
   81,  0,  0,  9;
   87, 39, 21,  0,  3;
   93,  0,  0,  0,  0;
   99, 45,  0,  0,  0;
  105,  0, 27, 15,  0;
  111, 51,  0,  0,  0;
  117,  0,  0,  0,  9;
  123, 57, 33,  0,  0,  3;
  129,  0,  0, 21,  0,  0;
  135, 63,  0,  0,  0,  0;
  141,  0, 39,  0,  0,  0;
  ...
For n = 9 the divisors of 9 are 1, 3, 9, therefore the sum of the divisors of 9 is 1 + 3 + 9 = 13 and 3*13 = 39. On the other hand the 9th row of triangle is 51, 21, 9, therefore the alternating row sum is 51 - 21 + 9 = 39, equaling 3 times sigma(9).
		

Crossrefs

Column 1 is A016945.

Cf. A000203, A001227, A003056, A074400, A196020, A236106, A237048, A237593, A239050, A239662, A244050, A272027.

Formula

T(n,k) = 3*A196020(n,k) = A196020(n,k) + A236106(n,k).