A108283 Triangle read by rows, generated from (..., 3, 2, 1).

1, 1, 3, 1, 5, 6, 1, 7, 17, 10, 1, 9, 34, 49, 15, 1, 11, 57, 142, 129, 21, 1, 13, 86, 313, 547, 321, 28, 1, 15, 121, 586, 1593, 2005, 769, 36, 1, 17, 162, 985, 3711, 7737, 7108, 1793, 45, 1, 19, 209, 1534, 7465, 22461, 36409, 24604, 4097, 55, 1, 21, 262, 2257, 13539, 54121, 131836, 167481, 83653, 9217, 66

Offset: 1

Author

Gary W. Adamson, May 30 2005

Keywords

• nonn
• tabl
• easy

Comments

Inverse binomial transforms of each column form the rows of A108284. Rightmost diagonal = triangular numbers, (A000217); while diagonals going to the left from (1, 3, 6, ...) are A000337 starting with 1: (1, 5, 17, 49, ...); A014915: (1, 7, 34, 142, ...); A014916: (1, 9, 57, ...); A014917: (1, 11, 86, ...).

Examples

			4th column = 10, 49, 142, 313, ... = f(x), x = 1, 2, 3; 4x^3 + 3x^2 + 2x + 1. f(3) = 142.
First few rows of the triangle:
  1;
  1,  3;
  1,  5,  6;
  1,  7, 17,  10;
  1,  9, 34,  49,  15;
  1, 11, 57, 142, 129, 21;
  ...
		

Crossrefs

Cf. A059045, A108284, A000217, A000337, A014915, A014916, A014917.

Programs

Maple

A108283 := proc(n,k)
    local x ;
    x := n-k+1 ;
    add( i*x^(i-1),i=1..k) ;
end proc:
seq(seq( A108283(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Sep 14 2016

Mathematica

T[, 1] := 1; T[n, n_] := n (n + 1)/2; T[n_, k_] := (1 - (n - k + 1)^k*(k^2 - k*n + 1))/(n - k)^2; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 13 2016 *)

Formula

n-th column = f(x), x = 1, 2, 3; n*x^(n-1) + (n-1)*x^(n-2) + (n-3)*x^(n-3) + ... + 1.

T(n,k) = (1+ (n-k+1)^k*(n*k-k^2-1))/ (n-k)^2, n>k. - Jean-François Alcover, Sep 13 2016

Extensions

More terms from Jean-François Alcover, Sep 13 2016