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

1,2

Comments

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

Links

Table of n, a(n) for n=1..105.

Formula

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

Example

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;
  ...

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 *)

Crossrefs

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

Sequence in context: A124846 A177375 A099512 * A130541 A128489 A034839

Adjacent sequences: A127093 A127094 A127095 * A127097 A127098 A127099

Keyword

nonn,easy,tabl

Author

Gary W. Adamson, Jan 05 2007

Extensions

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

Status

approved