A132774 A natural number operator.

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, 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, 0, 20, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 23

Offset

1,2

Comments

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

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

Links

Stefano Spezia, First 150 rows of the triangle, flattened

Formula

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

From Stefano Spezia, Dec 21 2021: (Start)

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

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

Example

First few rows of the triangle are:
  1;
  2,  3;
  0,  4,  5;
  0,  0,  6,  7;
  0,  0,  0,  8,  9;
  0,  0,  0,  0, 10, 11;
  ...

Mathematica

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 *)
Join[{1}, Flatten[{#, PadRight[{}, #[[1]]/2, 0]}&/@Partition[Range[2, 30], 2]]] (* Harvey P. Dale, Mar 24 2024 *)
Join[{1}, Flatten[Table[Join[Range[2n, 2n+1], PadRight[{}, n, 0]], {n, 20}]]] (* Harvey P. Dale, Mar 25 2024 *)

Crossrefs

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

Sequence in context: A175434 A154860 A284282 * A294721 A300816 A007945

Adjacent sequences: A132771 A132772 A132773 * A132775 A132776 A132777

Keyword

nonn,tabl

Author

Gary W. Adamson, Aug 28 2007

Extensions

More terms from Stefano Spezia, Dec 21 2021

Status

approved