|
%I #26 Jul 29 2019 12:25:45
%S 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,
%T 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,
%U 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
%N Triangle read by rows, A054533 * transpose(A101688) (matrix product) provided A101688 is read as a square array by antidiagonals upwards.
%C Right border = A000010, phi(n).
%C Row sums = A023896: (1, 1, 3, 4, 10, 6, 21, ...).
%H Jinyuan Wang, <a href="/A144734/b144734.txt">Table of n, a(n) for n = 1..10000</a>
%F 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).
%F From _Petros Hadjicostas_, Jul 28 2019: (Start)
%F 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.
%F T(n,k) = phi(n) - Sum_{d|n} d * mu(n/d) * ceiling(k/d) for n >= 2 and 1 <= k <= n.
%F (End)
%e First few rows of the triangle are as follows:
%e 1;
%e 0, 1;
%e 0, 1, 2;
%e 0, 0, 2, 2;
%e 0, 1, 2, 3, 4;
%e 0, -1, 0, 2, 3, 2;
%e 0, 1, 2, 3, 4, 5, 6;
%e 0, 0, 0, 0, 4, 4, 4, 4;
%e 0, 0, 0, 3, 3, 3, 6, 6, 6;
%e 0, -1, 0, -1, 0, 4, 5, 4, 5, 4;
%e 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
%e ...
%e row 4 = (0, 0, 2, 2) = partial sums from the right of row 4 of triangle A054533: (0, -2, 0, 2).
%Y Cf. A000010, A023896, A054533, A101688, A157658 (column 2).
%K tabl,sign
%O 1,6
%A _Gary W. Adamson_, Sep 20 2008
%E Name edited by and more terms from _Petros Hadjicostas_, Jul 28 2019
|
