A126615 Denominators in a harmonic triangle.
1, 2, 2, 2, 6, 3, 2, 6, 12, 4, 2, 6, 12, 20, 5, 2, 6, 12, 20, 30, 6, 2, 6, 12, 20, 30, 42, 7, 2, 6, 12, 20, 30, 42, 56, 8, 2, 6, 12, 20, 30, 42, 56, 72, 9, 2, 6, 12, 20, 30, 42, 56, 72, 90, 10, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 11, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 12, 2, 6
Offset: 1
Examples
Triangle T(n,k) begins: 1; 2, 2; 2, 6, 3; 2, 6, 12, 4; 2, 6, 12, 20, 5; 2, 6, 12, 20, 30, 6; 2, 6, 12, 20, 30, 42, 7; ... 1/1 = 1, 1/2 + 1/2 = 1, 1/2 + 1/6 + 1/3 = 1, 1/2 + 1/6 + 1/12 + 1/4 = 1, etc.
Programs
-
Maple
A126615 := (n,k) -> `if`(n=k, n, 1/Beta(k,2)); seq(print(seq(A126615(n,k), k=1..n)), n=1..9); # Peter Luschny, Jul 27 2014
-
Mathematica
Flatten@Table[{Array[2PolygonalNumber@#&,n],n+1},{n,0,10}] (* Giorgos Kalogeropoulos, Mar 31 2021 *)
Formula
Denominators of the inverse of A127949; numerators = 1. Triangle read by rows, first (n-1) terms of 1*2, 2*3, 3*4, ...; followed by "n".
T(n,k) = k*(k+1) = A002378(k) for k < n; T(n,n) = n. - Andrés Ventas, Mar 26 2021
Extensions
Gary W. Adamson submitted two different triangles numbered A127899 based on the harmonic numbers. This is the second of them, which I am renumbering as A126615. Unfortunately there were several other entries defined in terms of "A127899" and I may not have guessed which version of A127899 was being referred to. - N. J. A. Sloane, Jan 09 2007
More terms from Philippe Deléham, Dec 17 2008
Comments