A077051 Right summatory matrix, T, by antidiagonals.

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1

Offset: 1

Author

Clark Kimberling, Oct 22 2002

Keywords

• nonn
• tabl

Comments

If S=(s(1),s(2),...) is a sequence written as a row vector, then S*T is the summatory sequence of S; i.e. its n-th term is Sum{s(k): k|n}. T is the transpose of the left summatory matrix, A077049; T is the inverse of the right Moebius transformation matrix. See A077049 for further properties.

This is essentially the same as A051731, which includes only the triangle. Note that the standard in the OEIS is left to right antidiagonals, which would make this the left summatory matrix, and A077049 the right one. - Franklin T. Adams-Watters, Apr 08 2009

Sum of column k is A000005. - Geoffrey Critzer, Mar 29 2015

Examples

			Northwest corner:
1 1 1 1 1 1
0 1 0 1 0 1
0 0 1 0 0 1
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
		

Crossrefs

Cf. A077049, A077050, A077052.

Programs

Mathematica

(* returns the northwest corner *) nn = 20; Table[PadRight[Drop[CoefficientList[Series[x^n/(1 - x^n), {x, 0, nn}], x],1], nn], {n, 1, nn}] // Grid (* Geoffrey Critzer, Mar 29 2015 *)

Formula

T(n, k)=1 if n|k else T(n, k)=0.

From Boris Putievskiy, May 08 2013: (Start)

As table T(n,k)= floor(n/k)-floor((n-1)/k).

As linear sequence a(n) = floor(A002260(n)/A004736(n)) - floor((A002260(n)-1)/A004736(n));

a(n) = floor(i/j) - floor((i-1)/j), where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)

G.f. for row n: x^n/(1-x^n). - Geoffrey Critzer, Mar 29 2015