OEIS Unexpected Connections

[Paul D. Hanna]

Seqfans, 
     The magic of the OEIS is to uncover unexpected connections 
between seemingly unrelated methods of generating sequences. 
Herein is just one demonstration of this magic; of course, 
there are many other examples. 
 
A recent sequence submitted by Gary W. Adamson is intriguing. 
It is entry A125714 -- "Alfred Moessner's factorial triangle" 
I tried a variant of the generating rule, and found something 
quite unexpected. 
 
I encourage others to explore variations of the unusual recurrence 
of A125714 and find interesting versions for themselves! 
  
My variation is as follows. 
 
Start with a row of all 1's. Row n+1 equals the partial sums of row n 
excluding terms in columns k = {1,4,8,13,...} = m*(m+1)/2 - 2 for m>=1. 
The terms that are not to be included in the partial sums are 
enclosed in parenthesis in the table below. 
 
For example: 
row 2 = partial sums of [1, 3,4, 6,7,8, 10,11,12,13, ...]; 
row 3 = partial sums of [1, 8,14, 29,39,50, 75,90,106,123, ...];
row 4 = partial sums of [1, 23,52, 141,216,306, 535,695,876,1079, ...].
 
Rows of table begin: 
1,(1), 1, 1,(1), 1, 1, 1,(1), 1, 1, 1, 1,(1), 1, 1, 1, ...;
1,(2), 3, 4,(5), 6, 7, 8,(9), 10, 11, 12, 13,(14), 15, 16, 17, ...;
1,(4), 8, 14,(21), 29, 39, 50,(62), 75, 90, 106, 123,(141), 160, 181,.;
1,(9), 23, 52,(91), 141, 216, 306,(412), 535, 695, 876, 1079,(1305),..;
1,(24), 76, 217,(433), 739, 1274, 1969,(2845), 3924, 5479, 7335,...;
1,(77), 294, 1033,(2307), 4276, 8200, 13679,(21014), 30534, 45528,...;
1,(295), 1328, 5604,(13804), 27483, 58017, 103545,(167868), 255305,...;
1,(1329), 6933, 34416,(92433), 195978, 451283, 855463,(1454823),...;
1,(6934), 41350, 237328,(688611), 1544074, 3847960, 7700971,...;
1,(41351), 278679, 1822753,(5670713), 13371684, 35818351, 75299744,...;
...
  
The amazing connection here is this: 

column 1 equals A091352 = column 1 of triangle A091351, 
in which column k equals row sums of the matrix power A091351^k. 
 
I do not have a proof, but (at least) the initial 50 terms match exactly.

   
Triangle A091351 begins:
1;  
1, 1;  
1, 2, 1;  
1, 4, 3, 1;  
1, 9, 9, 4, 1;  
1, 24, 30, 16, 5, 1;  
1, 77, 115, 70, 25, 6, 1;  
1, 295, 510, 344, 135, 36, 7, 1;  
1, 1329, 2602, 1908, 805, 231, 49, 8, 1;  
1, 6934, 15133, 11904, 5325, 1616, 364, 64, 9, 1; ...  
 
where column k of A091351 = row sums of matrix power A091351^k for k>=0. 
  
Quite an unexepected connection between such diverse recurrences! 
   Paul 
 
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