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A295508 Triangle read by rows, related to binary partitions of n.

%I A295508 #6 Dec 01 2017 05:31:58
%S A295508 0,1,0,2,1,0,3,2,1,4,3,2,0,5,4,3,1,6,5,4,2,7,6,5,3,8,7,6,4,0,9,8,7,5,
%T A295508 1,10,9,8,6,2,11,10,9,7,3,12,11,10,8,4,13,12,11,9,5,14,13,12,10,6,15,
%U A295508 14,13,11,7,16,15,14,12,8,0,17,16,15,13,9,1
%N A295508 Triangle read by rows, related to binary partitions of n.
%F A295508 Let L(n) = (length of binary representation of n) - 0^n then
%F A295508 T(n, k) = n if k=0 else n - 2^(k-1) for n >= 0 and 0 <= k <= L(n).
%F A295508 Sum_{k=0..L(n)} T(n,k) = A123753(n-1) for n>=1.
%e A295508 0;
%e A295508 1,   0;
%e A295508 2,   1,  0;
%e A295508 3,   2,  1;
%e A295508 4,   3,  2,  0;
%e A295508 5,   4,  3,  1;
%e A295508 6,   5,  4,  2;
%e A295508 7,   6,  5,  3;
%e A295508 8,   7,  6,  4, 0;
%e A295508 9,   8,  7,  5, 1;
%e A295508 10,  9,  8,  6, 2;
%e A295508 11, 10,  9,  7, 3;
%e A295508 12, 11, 10,  8, 4;
%e A295508 13, 12, 11,  9, 5;
%e A295508 14, 13, 12, 10, 6;
%e A295508 15, 14, 13, 11, 7;
%p A295508 A295508_row := proc(n) local i, s, z; s := n; i := n-1; z := 1;
%p A295508 while 0 <= i do s := s,i; i := i-z; z := z+z od; s end:
%p A295508 seq(A295508_row(n), n=0..17);
%p A295508 # Alternatively after formula:
%p A295508 T := (n, k) -> `if`(k=0, n, n - 2^(k-1)):
%p A295508 L := n -> nops(convert(n, base, 2)) - 0^n:
%p A295508 T_row := n -> seq(T(n,k), k=0..L(n)):
%p A295508 seq(T_row(n), n=0..17);
%Y A295508 Cf. A123753.
%K A295508 nonn,tabf
%O A295508 0,4
%A A295508 _Peter Luschny_, Nov 30 2017