A127096 Triangle T(n,m) = A000012*A127094 read by rows.

1, 3, 1, 6, 1, 1, 10, 1, 3, 1, 15, 1, 3, 1, 1, 21, 1, 3, 4, 3, 1, 28, 1, 3, 4, 3, 1, 1, 36, 1, 3, 4, 7, 1, 3, 1, 45, 1, 3, 4, 7, 1, 6, 1, 1, 55, 1, 3, 4, 7, 6, 6, 1, 3, 1, 66, 1, 3, 4, 7, 6, 6, 1, 3, 1, 1, 78, 1, 3, 4, 7, 6, 12, 1, 7, 4, 3, 1, 91, 1, 3, 4, 7, 6, 12, 1, 7, 4, 3, 1, 1, 105, 1, 3, 4, 7, 6, 12, 8, 7, 4, 3, 1, 3, 1

Offset: 1

Author

Gary W. Adamson, Jan 05 2007

Keywords

• nonn
• easy
• tabl

Comments

Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127094 from the right.

Examples

			First few rows of the triangle are:
   1;
   3, 1,
   6, 1, 1;
  10, 1, 3, 1;
  15, 1, 3, 1, 1;
  21, 1, 3, 4, 3, 1;
  28, 1, 3, 4, 3, 1, 1;
  ...
		

Crossrefs

Cf. A127093, A127094, A123229, A024916 (row sums), A000203, A126988.

Programs

Maple

A127093 := proc(n,m) if n mod m = 0 then m; else 0 ; fi; end:
A127094 := proc(n,m) A127093(n, n-m+1) ; end:
A127096 := proc(n,m) add( A127094(j,m),j=m..n) ; end:
for n from 1 to 15 do for m from 1 to n do printf("%d,",A127096(n,m)) ; od: od: # R. J. Mathar, Aug 18 2009

Mathematica

T[n_, m_] := Sum[1 + Mod[j, m - j - 1] - Mod[1 + j, m - j - 1], {j, m, n}];
Table[T[n, m], {n, 1, 14}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 15 2023 *)

Formula

T(n,m) = Sum_{j=m..n} A000012(n,j)*A127094(j,m) = Sum_{j=m..n} A127094(j,m).

Extensions

Edited and extended by R. J. Mathar, Aug 18 2009