A130123 Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).

1, 0, 2, 0, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2048, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4096

Offset: 0

Author

Gary W. Adamson, May 11 2007

Keywords

• nonn
• tabl
• easy

Comments

A 2^n transform matrix.

A130123 * A007318 = A038208. A007318 * A130123 = A013609. A130124 = A130123 * A002260. A130125 = A128174 * A130123.

Triangle T(n,k), 0 <= k <= n, given by [0,0,0,0,0,0,...] DELTA [2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, May 26 2007

Also the Bell transform of A000038. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

T is the convolution triangle of the characteristic function of 2 (see A357368). - Peter Luschny, Oct 19 2022

Examples

			First few terms of the triangle:
  1;
  0, 2;
  0, 0, 4;
  0, 0, 0, 8;
  0, 0, 0, 0, 16;
  0, 0, 0, 0,  0, 32; ...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A130124, A130125.

Programs

Magma

[[k eq n select 2^n else 0: k in [0..n]]: n in [0..14]]; // G. C. Greubel, Jun 05 2019

Maple

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n=0,2,0), 9); # Peter Luschny, Jan 27 2016
# Uses function PMatrix from A357368.
PMatrix(10, n -> ifelse(n=1, 2, 0)); # Peter Luschny, Oct 19 2022

Mathematica

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# == 0, 2, 0]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
Table[If[k==n, 2^n, 0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 05 2019 *)

PARI

{T(n,k) = if(k==n, 2^n, 0)}; \\ G. C. Greubel, Jun 05 2019

Sage

def T(n, k):
    if (k==n): return 2^n
    else: return 0
[[T(n, k) for k in (0..n)] for n in (0..14)] # G. C. Greubel, Jun 05 2019

Formula

G.f.: 1/(1-2*x*y). - R. J. Mathar, Aug 11 2015