A089000 Square table, read by antidiagonals, of coefficients T(k,n) (row k; column n) defined by: T(k,n) = k*T(k,n-1)+ n; T(k,0) = 0.

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 6, 4, 1, 0, 5, 10, 11, 5, 1, 0, 6, 15, 26, 18, 6, 1, 0, 7, 21, 57, 58, 27, 7, 1, 0, 8, 28, 120, 179, 112, 38, 8, 1, 0, 9, 36, 247, 543, 453, 194, 51, 9, 1, 0, 10, 45, 502, 1636, 1818, 975, 310, 66, 10, 1, 0

Offset: 0

Author

Philippe Deléham, Nov 02 2003

Keywords

• easy
• nonn
• tabl

Crossrefs

Rows begin:

{0, 1, 2, 3, 4, 5, 6, 7, 8, ...}:see A001477

{0, 1, 3, 6, 10, 15, 21, 28, ...} : see A000217

{0, 1, 4, 11, 26, 57, 120, 247, 502, ...} : see A000295

{0, 1, 5, 18, 58, 179, 543, 1636, ...} : see A000340

{0, 1, 6, 27, 112, 453, 1818, 7279, ...} : see A014825

{0, 1, 7, 38, 194, 975, 4881, 24412, ...} : see A014827

{0, 1, 8, 51, 310, 1865, 11196, 67183, ...}: see diagonals of triangle A088990

Diagonal begin:

{0, 1, 4, 18, 112, 975, 11196, ... } :see A062805

{0, 1, 5, 27, 194, 1865, ...} : see A023811

Column {3, 6, 11, 18, 27, 38, 51, ...} : see A010000

Programs

Mathematica

Unprotect[Power]; 0^0=1; T[n_,k_]:=Sum[j*k^(n-j),{j,0,n}]; Table[T[n-k,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Apr 19 2025 *)

Formula

T(k, n)= (k^(n+1)- (k-1)*n - k)/(k-1)^2. T(k, n) = Sum(j, 0<=j<=n; j*k^(n-j)).