A144734 Triangle read by rows, A054533 * transpose(A101688) (matrix product) provided A101688 is read as a square array by antidiagonals upwards.

1, 0, 1, 0, 1, 2, 0, 0, 2, 2, 0, 1, 2, 3, 4, 0, -1, 0, 2, 3, 2, 0, 1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 3, 3, 3, 6, 6, 6, 0, -1, 0, -1, 0, 4, 5, 4, 5, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 0, -2, -2, 0, 0, 4, 4, 6, 6, 4, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, -1, 0

Offset

1,6

Comments

Right border = A000010, phi(n).

Row sums = A023896: (1, 1, 3, 4, 10, 6, 21, ...).

Links

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

Formula

Triangle read by rows, A054533 * transpose(A101688) (matrix product); i.e., partial sums from of the right of triangle A054533 (because A101688 can be viewed as an upper triangular matrix of 1's).

From Petros Hadjicostas, Jul 28 2019: (Start)

T(n,k) = Sum_{m = k..n} A054533(n,m) = Sum_{d|n} d * mu(n/d) * ((n/d) - ceiling(k/d) + 1) for n >= 1 and 1 <= k <= n.

T(n,k) = phi(n) - Sum_{d|n} d * mu(n/d) * ceiling(k/d) for n >= 2 and 1 <= k <= n.

(End)

Example

First few rows of the triangle are as follows:
  1;
  0,  1;
  0,  1,  2;
  0,  0,  2,  2;
  0,  1,  2,  3,  4;
  0, -1,  0,  2,  3,  2;
  0,  1,  2,  3,  4,  5,  6;
  0,  0,  0,  0,  4,  4,  4,  4;
  0,  0,  0,  3,  3,  3,  6,  6,  6;
  0, -1,  0, -1,  0,  4,  5,  4,  5,  4;
  0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10;
  ...
row 4 = (0, 0, 2, 2) = partial sums from the right of row 4 of triangle A054533: (0, -2, 0, 2).

Crossrefs

Cf. A000010, A023896, A054533, A101688, A157658 (column 2).

Sequence in context: A230805 A230994 A337509 * A029361 A275966 A284059

Adjacent sequences: A144731 A144732 A144733 * A144735 A144736 A144737

Keyword

tabl,sign

Author

Gary W. Adamson, Sep 20 2008

Extensions

Name edited by and more terms from Petros Hadjicostas, Jul 28 2019

Status

approved