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%I #22 Oct 19 2022 10:56:16
%S 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,
%T 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,
%U 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
%N 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, ...).
%C A 2^n transform matrix.
%C A130123 * A007318 = A038208. A007318 * A130123 = A013609. A130124 = A130123 * A002260. A130125 = A128174 * A130123.
%C 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
%C Also the Bell transform of A000038. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%C T is the convolution triangle of the characteristic function of 2 (see A357368). - _Peter Luschny_, Oct 19 2022
%H G. C. Greubel, <a href="/A130123/b130123.txt">Rows n = 0..100 of triangle, flattened</a>
%F G.f.: 1/(1-2*x*y). - _R. J. Mathar_, Aug 11 2015
%e First few terms of the triangle:
%e 1;
%e 0, 2;
%e 0, 0, 4;
%e 0, 0, 0, 8;
%e 0, 0, 0, 0, 16;
%e 0, 0, 0, 0, 0, 32; ...
%p # The function BellMatrix is defined in A264428.
%p BellMatrix(n -> `if`(n=0,2,0), 9); # _Peter Luschny_, Jan 27 2016
%p # Uses function PMatrix from A357368.
%p PMatrix(10, n -> ifelse(n=1, 2, 0)); # _Peter Luschny_, Oct 19 2022
%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t rows = 12;
%t M = BellMatrix[If[# == 0, 2, 0]&, rows];
%t Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)
%t Table[If[k==n, 2^n, 0], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Jun 05 2019 *)
%o (PARI) {T(n,k) = if(k==n, 2^n, 0)}; \\ _G. C. Greubel_, Jun 05 2019
%o (Magma) [[k eq n select 2^n else 0: k in [0..n]]: n in [0..14]]; // _G. C. Greubel_, Jun 05 2019
%o (Sage)
%o def T(n, k):
%o if (k==n): return 2^n
%o else: return 0
%o [[T(n, k) for k in (0..n)] for n in (0..14)] # _G. C. Greubel_, Jun 05 2019
%Y Cf. A130124, A130125.
%K nonn,tabl,easy
%O 0,3
%A _Gary W. Adamson_, May 11 2007
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