login
A natural number operator.
3

%I #23 Mar 25 2024 15:03:06

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

%T 0,0,14,15,0,0,0,0,0,0,0,16,17,0,0,0,0,0,0,0,0,18,19,0,0,0,0,0,0,0,0,

%U 0,20,21,0,0,0,0,0,0,0,0,0,0,22,23

%N A natural number operator.

%C Row sums = A016813: (1, 5, 9, 13, ...).

%C A132774 * [1, 2, 3, ...] = A033951.

%H Stefano Spezia, <a href="/A132774/b132774.txt">First 150 rows of the triangle, flattened</a>

%F As an infinite lower triangular matrix, (1, 3, 5, ...) in the main diagonal and (2, 4, 6, ...) in the subdiagonal; with the rest zeros.

%F From _Stefano Spezia_, Dec 21 2021: (Start)

%F T(n, k) = 2*n - 1 if n = k, T(n, k) = 2*(n - 1) if n - k = 1, otherwise T(n, k) = 0.

%F G.f.: x*y*(1 + x*(2 + y))/(1 - x*y)^2. (End)

%e First few rows of the triangle are:

%e 1;

%e 2, 3;

%e 0, 4, 5;

%e 0, 0, 6, 7;

%e 0, 0, 0, 8, 9;

%e 0, 0, 0, 0, 10, 11;

%e ...

%t T[n_,k_]:=If[n==k,2n-1,If[n-k==1,2(n-1),0]]; Flatten[Table[T[n,k],{n,12},{k,n}]] (* _Stefano Spezia_, Dec 21 2021 *)

%t Join[{1},Flatten[{#,PadRight[{},#[[1]]/2,0]}&/@Partition[Range[2,30],2]]] (* _Harvey P. Dale_, Mar 24 2024 *)

%t Join[{1},Flatten[Table[Join[Range[2n,2n+1],PadRight[{},n,0]],{n,20}]]] (* _Harvey P. Dale_, Mar 25 2024 *)

%Y Cf. A016813 (row sums), A033951, A060747 (main diagonal).

%K nonn,tabl

%O 1,2

%A _Gary W. Adamson_, Aug 28 2007

%E More terms from _Stefano Spezia_, Dec 21 2021