Knuth's Power Tree and Related Sequences

[Paul D. Hanna]

Seqfans, 
      Would someone like to compute more terms of Knuth's Power Tree? 
It apparently is NOT in the OEIS, but seems that it should be. 
The terms below I simply copied from the book (see reference below), 
so I hope that I do not have a typo. 
  
I am mostly interested in knowing these 2 related sequences: 
* number of nodes in rows:   1,1,2,3,5,9,15,26,... 
* row sums:   1,2,7,19,54,164,476,1417,... 
but am having difficulty computing the above. 
 
Can anyone help generate these 3 new sequences? 
 
Thanks, 
       Paul 
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SEQUENCE:
1,2,3,4,5,6,8,7,10,9,12,16,14,11,13,15,20,18,24,17,32,19,21,
28,22,23,26,25,30,40,27,36,48,33,34,64,38,35,42,29,31,56,44,
46,39,52,50,45,60,41,43,80,54,37,72,49,51,96,66,68,65,128,
 
NAME:
The power tree (as defined by Knuth), read by rows, 
where T(n,k) is the label of the k-th node in row n. 
 
COMMENTS:
The power tree is generated by a systematic method that is supposed 
to give the minimum number of multiplications to arrive at x^n. 
 
REFERENCES: 
Donald E. Knuth, The Art of Computer Programming Third Edition. Vol. 2, 
Seminumerical Algorithms. Chapter 4.6.3 Evaluation of Powers, Page 464. 
Addison-Wesley, Reading, MA, 1997.  
 
FORMULA:
Start the root node with label (1) in row 1. 
Assuming that the first n rows have been constructed, 
define row (n+1) of the power tree as follows. 
Proceeding from left to right in row n of the tree, 
take each node labelled L = T(n,k) and let the labels 
[1, T(2,j_2), T(3,j_3), ..., T(n,j_n)], where j_n=k, 
form the path from the root of the tree to node T(n,k). 
Attach to node (L) new nodes with labels generated by: 
[L+1, L+T(2,j_2), L+T(3,j_3), ..., L+T(n,k)=2*L] 
after discarding any label that has appeared before in the tree. 
 
EXAMPLE:
The rows of the power tree begin: 
1; 
2; 
3,4; 
5,6,8; 
7,10,9,12,16; 
14,11,13,15,20,18,24,17,32; 
19,21,28,22,23,26,25,30,40,27,36,48,33,34,64; 
38,35,42,29,31,56,44,46,39,52,50,45,60,41,43,80,54,37,72,49,51,96,66,68,6
5,128; 
where nodes are attached to each other like so: 
1->[2]; 
2->[3,4]; 
3->[5,6], 4->[8]; 
5->[7,10], 6->[9,12], 8->[16]; 
7->[14], 10->[11,13,15,20], 9->[18], 12->[24], 16->[32]; ... 
Ex., the path from root node (1) to node (10) is [1,2,3,5,10], 
so the possible labels for nodes to be attached to node (10) are: 
[10+1,10+2,10+3,10+5,10+10]; but label (12) has already been used, 
so 4 nodes with labels [11,13,15,20] are attached to node (10). 
 
SEE ALSO:
Cf. A______ (number of nodes in rows), A______ (row sums).
 
OFFSET: 1