A236106 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the twice odd numbers (A016825) interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

2, 6, 10, 2, 14, 0, 18, 6, 22, 0, 2, 26, 10, 0, 30, 0, 0, 34, 14, 6, 38, 0, 0, 2, 42, 18, 0, 0, 46, 0, 10, 0, 50, 22, 0, 0, 54, 0, 0, 6, 58, 26, 14, 0, 2, 62, 0, 0, 0, 0, 66, 30, 0, 0, 0, 70, 0, 18, 10, 0, 74, 34, 0, 0, 0, 78, 0, 0, 0, 6, 82, 38, 22, 0, 0, 2

Offset: 1

Author

Omar E. Pol, Jan 23 2014

Keywords

• nonn
• tabf

Comments

Gives an identity for the twice sigma function (A074400), the sum of the even divisors of 2n.

Alternating sum of row n equals A074400(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = 2*A000203(n) = A074400(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).

For more information see A196020.

Examples

			Triangle begins:
  2;
  6;
  10,  2;
  14,  0;
  18,  6;
  22,  0,  2;
  26, 10,  0;
  30,  0,  0;
  34, 14,  6;
  38,  0,  0,  2;
  42, 18,  0,  0;
  46,  0, 10,  0;
  50, 22,  0,  0;
  54,  0,  0,  6;
  58, 26, 14,  0,  2;
  62,  0,  0,  0,  0;
  66, 30,  0,  0,  0;
  70,  0, 18, 10,  0;
  74, 34,  0,  0,  0;
  78,  0,  0,  0,  6;
  82, 38, 22,  0,  0,  2;
  86,  0,  0, 14,  0,  0;
  90, 42,  0,  0,  0,  0;
  94,  0, 26,  0,  0,  0;
  ...
For n = 9 the divisors of 2*9 = 18 are 1, 2, 3, 6, 9, 18, therefore the sum of the even divisors of 18 is 2 + 6 + 18 = 26. On the other hand the 9th row of triangle is 34, 14, 6, therefore the alternating row sum is 34 - 14 + 6 = 26, equaling the sum of the even divisors of 18.
If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of the even divisors of 2n. Example: for n = 12 the sum of the even divisors of 2*12 = 24 is 2 + 4 + 6 + 8 + 12 + 24 = 56, and the alternating sum of the 12th row of triangle is 46 - 0 + 10 - 0 = 56.
		

Crossrefs

Cf. A000203, A000217, A001227, A003056, A016825, A074400, A196020, A211343, A228813, A231345, A231347, A235791, A235794, A236104, A236112.

Formula

T(n,k) = 2*A196020(n,k).