A129186 Right shift operator generating 1's in shifted spaces.

1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Gary W. Adamson, Apr 01 2007

Keywords

• nonn
• easy
• tabl

Comments

Let A129186 = M, then M*V, V a vector; shifts V to the right, appending 1's to the shifted spaces. Example: M*V, V = [1,2,3,...] = [1,1,2,3,...].

Triangle T(n,k), read by rows, given by (1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 08 2011

Examples

			First few rows of the triangle are:
1;
1, 0;
0, 1, 0;
0, 0, 1, 0;
0, 0, 0, 1, 0;
...
		

Links

• Robert Price, Table of n, a(n) for n = 0..5049

Crossrefs

Generalized Eulerian triangles: this sequence (m=0), A173018 (m=1), A292604 (m=2).

Cf. A000012 (row sums), A071919, A129184, A129185.

Programs

Maple

gf := 1 + z/(1 - x*z): ser := series(gf, z, 16): c := k -> coeff(ser, z, k):
seq(seq(coeff(c(n), x, k), k=0..n), n=0..14); # Peter Luschny, Jul 07 2019

Mathematica

Join[{1},Flatten[Table[PadLeft[{1,0},n,0],{n,2,20}]]] (* Harvey P. Dale, Aug 26 2019 *)

Formula

Infinite lower triangular matrix with (1,0,0,...) in the main diagonal and (1,1,1...) in the subdiagonal.

G.f.: (1-(y-1)*x)/(1-y*x). - Philippe Deléham, Dec 08 2011