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

0,6

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).

Links

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

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)

Example

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;

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A209777 A356931 A356864 * A128229 A132013 A105820

Adjacent sequences: A145674 A145675 A145676 * A145678 A145679 A145680

Keyword

nonn,tabl,easy

Author

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

Extensions

Edited by R. J. Mathar, Oct 02 2009

Status

approved