A144824 Triangle read by rows, A054533 * A127648 (matrix product).

1, -1, 2, -1, -2, 6, 0, -4, 0, 8, -1, -2, -3, -4, 20, 1, -2, -6, -4, 5, 12, -1, -2, -3, -4, -5, -6, 42, 0, 0, 0, -16, 0, 0, 0, 32, 0, 0, -9, 0, 0, -18, 0, 0, 54, 1, -2, 3, -4, -20, -6, 7, -8, 9, 40, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, 110

Offset: 1

Author

Gary W. Adamson, Sep 21 2008

Keywords

• tabl
• sign

Comments

Row sums = A023896: (1, 1, 3, 4, 10, 16, 21, ...).

Right border = A002618: (1, 2, 6, 8, 20, 12, ...).

Left border = mu(n) = A008683 (n).

Examples

			Triangle A054533 starts as follows:
   1;
  -1,  1;
  -1, -1,  2;
   0, -2,  0,  2;
  -1, -1, -1, -1, 4;
   1, -1, -2, -1, 1, 2;
   ...
The first few rows of triangle A144824 are as follows:
   1;
  -1,  2;
  -1, -2,  6;
   0, -4,  0,  8;
  -1, -2, -3, -4, 20;
   1, -2, -6, -4,  5, 12;
  -1, -2, -3, -4, -5, -6, 42;
   ...
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002260, A002618, A008683, A023896, A054533, A127648.

Formula

Triangle read by rows, A054533 * A127648 (matrix product). The operation is equivalent to taking termwise products of row A054533 terms and the natural numbers.

T(n, k) = k * Sum_{d|gcd(n,k)} d * mu(n/d) for n >= 1 and 1 <= k <= n. - Petros Hadjicostas, Jul 28 2019

a(n) = A002260(n)*A054533(n). - Jinyuan Wang, Jul 29 2019