A129234 Triangle read by rows: T(n,k) = n/k + k - 1 if n mod k = 0; otherwise T(n,k)=0 (1 <= k <= n).

1, 2, 2, 3, 0, 3, 4, 3, 0, 4, 5, 0, 0, 0, 5, 6, 4, 4, 0, 0, 6, 7, 0, 0, 0, 0, 0, 7, 8, 5, 0, 5, 0, 0, 0, 8, 9, 0, 5, 0, 0, 0, 0, 0, 9, 10, 6, 0, 0, 6, 0, 0, 0, 0, 10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 12, 7, 6, 6, 0, 7, 0, 0, 0, 0, 0, 12, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 14, 8, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Author

Gary W. Adamson, Apr 05 2007

Keywords

• nonn
• tabl

Comments

Row sums = A129235: (1, 4, 6, 11, 10, 20, 14, ...). Moebius transform of A129234 = A129236. Inverse Moebius transform of A129234 = A129237.

Examples

			First few rows of the triangle:
  1;
  2, 2;
  3, 0, 3;
  4, 3, 0, 4;
  5, 0, 0, 0, 5;
  6, 4, 4, 0, 0, 6;
  7, 0, 0, 0, 0, 0, 7;
  ...
		

Crossrefs

Cf. A129235, A129236, A129237.

Programs

Maple

T:=proc(n,k) if n mod k = 0 then n/k+k-1 else 0 fi end: for n from 1 to 16 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Apr 17 2007

Mathematica

T[n_,k_]:=If[Mod[n,k]==0,n/k+k-1,0];Table[T[n,k],{n,14},{k,n}]//Flatten (* James C. McMahon, Jan 17 2025 *)

PARI

row(n) = vector(n, k, if (!(n%k), n/k + k - 1, 0)); \\ Michel Marcus, Jan 17 2025

Formula

G.f. = G(t,z) = Sum_{k>=1} t^k*z^k*(k-(k-1)*z^k)/(1-z^k)^2. - Emeric Deutsch, Apr 17 2007

Extensions

Edited by Emeric Deutsch, Apr 17 2007