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A128307 Triangle, (1, 0, 1, 2, 4, 8, ...) in every column.

%I A128307 #17 Feb 08 2022 19:25:21
%S A128307 1,0,1,1,0,1,2,1,0,1,4,2,1,0,1,8,4,2,1,0,1,16,8,4,2,1,0,1,32,16,8,4,2,
%T A128307 1,0,1,64,32,16,8,4,2,1,0,1,128,64,32,16,8,4,2,1,0,1,256,128,64,32,16,
%U A128307 8,4,2,1,0,1,512,256,128,64,32,16,8,4,2,1,0
%N A128307 Triangle, (1, 0, 1, 2, 4, 8, ...) in every column.
%C A128307 Row sums = (1, 1, 2, 4, 8, ...). A128308 = binomial transform of A128307.
%C A128307 Riordan array ( 1 + x^2/(1 - 2*x), x ). T(n,k) gives the number of compositions of n of the form 1 + 1 + ... + 1 + a_1 + ... + a_m beginning with k 1's and with a_1 > 1. See Shapiro, Section 5. An example is given below. - _Peter Bala_, Aug 18 2014
%H A128307 Harvey P. Dale, <a href="/A128307/b128307.txt">Table of n, a(n) for n = 1..1000</a>
%H A128307 L. Shapiro, <a href="https://www.semanticscholar.org/paper/A-Survey-of-the-Riordan-Group-Getie/94e657b0dc55bd38408080fd86189ae8bcf0ec8a">A survey of the Riordan group</a>
%F A128307 (1, 0, 1, 2, 4, 8, ...) in every column.
%e A128307 First few rows of the triangle:
%e A128307   1;
%e A128307   0, 1;
%e A128307   1, 0, 1;
%e A128307   2, 1, 0, 1;
%e A128307   4, 2, 1, 0, 1;
%e A128307   8, 4, 2, 1, 0, 1;
%e A128307   ...
%e A128307 From _Peter Bala_, Aug 18 2014: (Start)
%e A128307 Row 4: [4,2,1,0,1]
%e A128307               Compositions                Number
%e A128307 k = 0     4, 3 + 1, 2 + 2, 2 + 1 + 1        4
%e A128307 k = 1     1 + 3, 1 + 2 + 1                  2
%e A128307 k = 2     1 + 1 + 2                         1
%e A128307 k = 3                                       0
%e A128307 k = 4     1 + 1 + 1 + 1                     1
%e A128307 (End)
%t A128307 Join[{1,0,1},Table[Join[NestWhileList[#/2&,2^n,#!=1&],{0,1}],{n,0,10}]]//Flatten (* _Harvey P. Dale_, Nov 25 2018 *)
%Y A128307 Cf. A007318, A128308.
%K A128307 nonn,tabl
%O A128307 1,7
%A A128307 _Gary W. Adamson_, Feb 25 2007
%E A128307 More terms from _Harvey P. Dale_, Nov 25 2018