A176702 -id:A176702 - cp's OEIS frontend

A134541 Triangle read by rows: A000012 * A054525 regarded as infinite lower triangular matrices.

1, 0, 1, -1, 1, 1, -1, 0, 1, 1, -2, 0, 1, 1, 1, -1, -1, 0, 1, 1, 1, -2, -1, 0, 1, 1, 1, 1, -2, -1, 0, 0, 1, 1, 1, 1, -2, -1, -1, 0, 1, 1, 1, 1, 1, -1, -2, -1, 0, 0, 1, 1, 1, 1, 1, -2, -2, -1, 0, 0, 1, 1, 1, 1, 1, 1, -2, -1, -1, -1, 0, 0, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Oct 31 2007

Row sums = 1.
Left border = A002321, the Mertens function.
A134541 * [1,2,3,...] = A002088: (1, 2, 4, 6, 10, 12, 18, 22, ...).

			First few rows of the triangle:
   1;
   0,  1;
  -1,  1,  1;
  -1,  0,  1, 1;
  -2,  0,  1, 1, 1;
  -1, -1,  0, 1, 1, 1;
  -2, -1,  0, 1, 1, 1, 1;
  -2, -1,  0, 0, 1, 1, 1, 1;
  -2, -1, -1, 0, 1, 1, 1, 1, 1;
  -1, -2, -1, 0, 0, 1, 1, 1, 1, 1;
  ...

Cf. A000012, A054525, A002321, A002088, A134541.

Matrix inverse of A176702. - Mats Granvik, Apr 24 2010

Clear[t, s, n, k, z, x]; z = 1; nn = 10; t[n_, k_] := t[n, k] = If[n >= k, If[k == 1, 1 - Sum[t[n, k + i]/(i + 1)^(s - 1), {i, 1, n - 1}], t[Floor[n/k], 1]], 0]; Flatten[Table[Table[Limit[t[n, k], s -> z], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Jul 22 2012 *) (* updated Mats Granvik, Apr 10 2016 *)

Recurrence: T(n, k) = If n >= k then If k = 1 then 1 - Sum_{i=1..n-1} T(n, k + i)/(i + 1)^(s - 1) else T(floor(n/k) else 1)) else 0). - Mats Granvik, Apr 17 2016

More terms from Amiram Eldar, Jun 09 2024

A180430 Triangle T(n,k) defined by the recurrence in the formula.

Original entry on oeis.org

1, 1, 2, 1, 0, 2, 1, 2, -2, 2, 1, 0, 2, -2, 2, 1, 2, 0, 0, -2, 2, 1, 0, 0, 2, 0, -2, 2, 1, 2, 0, 0, 0, 0, -2, 2, 1, 0, 2, -2, 2, 0, 0, -2, 2, 1, 2, -2, 2, 0, 0, 0, 0, -2, 2, 1, 0, 2, 0, -2, 2, 0, 0, 0, -2, 2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 1, 0, 0, 0, 2, -2, 2, 0, 0, 0, 0, -2, 2, 1, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Sep 04 2010

			Table starts:
  1;
  1,  2;
  1,  0,  2;
  1,  2, -2,  2;
  1,  0,  2, -2,  2;
  1,  2,  0,  0, -2,  2;
  1,  0,  0,  2,  0, -2,  2;
  1,  2,  0,  0,  0,  0, -2,  2;
  1,  0,  2, -2,  2,  0,  0, -2,  2;
  1,  2, -2,  2,  0,  0,  0,  0, -2,  2;

Cf. A176702.

Using European dot comma style:
=if(column()=1; 1; if(row()>=column(); sum(indirect(address(row()-column()+2; column()-1; 4)&":"&address(row()-1; column()-1; 4); 4))-sum(indirect(address(row()-column()+1; column(); 4)&":"&address(row()-1; column(); 4); 4)); 0))

A180430 := proc(n,k) option remember; if k>n then 0; elif k = 1 then 1; elif k = 2 then 2-procname(n-1,k) ; else add( procname(n-i,k-1),i=1..k-2) - add(procname(n-i,k),i=1..k-1) ; end if; end proc: # R. J. Mathar, Jul 13 2011

T(n,1)=1. T(n,2) = 2-T(n-1,2). T(n,k) = Sum_{i=1..k-2} T(n-i,k-1) - Sum{i=1..k-1} T(n-i,k) for k > 2.

cp's OEIS Frontend

A134541 Triangle read by rows: A000012 * A054525 regarded as infinite lower triangular matrices.

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