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

1,3

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, ...).

Links

Table of n, a(n) for n=1..66.

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

Example

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;
  ...

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 *)

Crossrefs

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

Sequence in context: A134083 A210551 A113445 * A208904 A344479 A209754

Adjacent sequences: A108280 A108281 A108282 * A108284 A108285 A108286

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, May 30 2005

Extensions

More terms from Jean-François Alcover, Sep 13 2016

Status

approved