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A132774 A natural number operator.

%I A132774 #23 Mar 25 2024 15:03:06
%S A132774 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 A132774 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 A132774 0,20,21,0,0,0,0,0,0,0,0,0,0,22,23
%N A132774 A natural number operator.
%C A132774 Row sums = A016813: (1, 5, 9, 13, ...).
%C A132774 A132774 * [1, 2, 3, ...] = A033951.
%H A132774 Stefano Spezia, <a href="/A132774/b132774.txt">First 150 rows of the triangle, flattened</a>
%F A132774 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 A132774 From _Stefano Spezia_, Dec 21 2021: (Start)
%F A132774 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 A132774 G.f.: x*y*(1 + x*(2 + y))/(1 - x*y)^2. (End)
%e A132774 First few rows of the triangle are:
%e A132774   1;
%e A132774   2,  3;
%e A132774   0,  4,  5;
%e A132774   0,  0,  6,  7;
%e A132774   0,  0,  0,  8,  9;
%e A132774   0,  0,  0,  0, 10, 11;
%e A132774   ...
%t A132774 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 A132774 Join[{1},Flatten[{#,PadRight[{},#[[1]]/2,0]}&/@Partition[Range[2,30],2]]] (* _Harvey P. Dale_, Mar 24 2024 *)
%t A132774 Join[{1},Flatten[Table[Join[Range[2n,2n+1],PadRight[{},n,0]],{n,20}]]] (* _Harvey P. Dale_, Mar 25 2024 *)
%Y A132774 Cf. A016813 (row sums), A033951, A060747 (main diagonal).
%K A132774 nonn,tabl
%O A132774 1,2
%A A132774 _Gary W. Adamson_, Aug 28 2007
%E A132774 More terms from _Stefano Spezia_, Dec 21 2021