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%I #27 Jan 04 2024 13:23:23
%S 1,0,1,0,0,2,0,0,0,4,0,0,0,0,8,0,0,0,0,0,16,0,0,0,0,0,0,32,0,0,0,0,0,
%T 0,0,64,0,0,0,0,0,0,0,0,128,0,0,0,0,0,0,0,0,0,256,0,0,0,0,0,0,0,0,0,0,
%U 512,0,0,0,0,0,0,0,0,0,0,0,1024,0,0,0,0,0,0,0,0,0,0,0,0,2048,0,0,0,0,0,0,0
%N Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).
%C As infinite lower triangular matrices, binomial transform of A134309 = A082137. A134309 * A007318 = A055372. A134309 * [1,2,3,...] = A057711: (1, 2, 6, 16, 40, 96, 224,...).
%C Triangle read by rows given by [0,0,0,0,0,0,0,0,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 20 2007
%F Triangle, T(0,0) = 1, then for n > 0, n zeros followed by 2^(n-1). Infinite lower triangular matrix with (1, 1, 2, 4, 8, 16, ...) in the main diagonal and the rest zeros.
%F G.f.: (1 - y*x)/(1 - 2*y*x). - _Philippe Deléham_, Feb 04 2012
%F Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = 0, 1, 2, 3, 4, 5, 6 respectively. - _Philippe Deléham_, Feb 04 2012
%F Diagonal is A011782, other elements are 0. - _M. F. Hasler_, Mar 29 2022
%e Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e 1;
%e 0, 1;
%e 0, 0, 2;
%e 0, 0, 0, 4;
%e 0, 0, 0, 0, 8;
%e 0, 0, 0, 0, 0, 16;
%e ...
%t Join[{1},Flatten[Table[Join[{PadRight[{},n],2^(n-1)}],{n,20}]]] (* _Harvey P. Dale_, Jan 04 2024 *)
%o (PARI) A134309(r,c)=if(r==c,2^max(r-1,0),0) \\ _M. F. Hasler_, Mar 29 2022
%Y Cf. A011782 (diagonal elements: 1 followed by 1, 2, 4, 8, ... = A000079: 2^n).
%Y Cf. A055372, A057711, A082137.
%K nonn,tabl
%O 0,6
%A _Gary W. Adamson_, Oct 19 2007
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