A145677 Triangle T(n, k) read by rows: T(n, 0) = 1, T(n, n) = n, n>0, T(n,k) = 0, 0 < k < n-1.

1, 1, 1, 1, 0, 2, 1, 0, 0, 3, 1, 0, 0, 0, 4, 1, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Mar 28 2009

Keywords

• nonn
• tabl
• easy

Comments

The first entry in each row is 1, the last entry in each of the rows consist of the positive integers (starting 1,1,2,3,...), and all other entries in the triangle are 0's (see example).

The vector of (1, 1, 2, 5, 16, 65, 326,...), which is 1 followed by A000522, is an eigenvector of the matrix: 1 + Sum_{k=1..n} T(n,k)*A000522(k-1) = A000522(n).

Examples

			First few rows of the triangle:
  1;
  1, 1;
  1, 0, 2;
  1, 0, 0, 3;
  1, 0, 0, 0, 4;
  1, 0, 0, 0, 0, 5;
  1, 0, 0, 0, 0, 0, 6;
  1, 0, 0, 0, 0, 0, 0, 7;
  1, 0, 0, 0, 0, 0, 0, 0, 8;
  1, 0, 0, 0, 0, 0, 0, 0, 0, 9;
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A128229, A002061, A000522, A152271, A158416.

Programs

Mathematica

T[n_, k_]:= If[k==0, 1, If[k==n, n, 0]];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 23 2021 *)

Sage

def A145677(n,k):
    if (k==0): return 1
    elif (k==n): return n
    else: return 0
flatten([[A145677(n,k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Dec 23 2021

Formula

T(n, k) = A158821(n,n-k).

1 + Sum_{k= 1..n} T(n,k) *(k-1) = A002061(n).

From G. C. Greubel, Dec 23 2021: (Start)

Sum_{k=0..n} T(n, k) = A000027(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A158416(n) = A152271(n+1). (End)

Extensions

Edited by R. J. Mathar, Oct 02 2009