VOOZH about

URL: https://oeis.org/A130123

⇱ A130123 - OEIS

login
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, ...).
8
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
0,3
COMMENTS
A 2^n transform matrix.
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
FORMULA
G.f.: 1/(1-2*x*y). - R. J. Mathar, Aug 11 2015
EXAMPLE
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; ...
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 *)
PROG
(PARI) {T(n, k) = if(k==n, 2^n, 0)}; \\ G. C. Greubel, Jun 05 2019
(Magma) [[k eq n select 2^n else 0: k in [0..n]]: n in [0..14]]; // G. C. Greubel, Jun 05 2019
(SageMath)
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
CROSSREFS
Sequence in context: A028590 A074644 A321256 * A319935 A136337 A028601
KEYWORD
nonn,tabl,easy
AUTHOR
Gary W. Adamson, May 11 2007
STATUS
approved