[seqfan] Quest of a trinagular decomposition of A081113

[Richard Mathar]

I received a request to provide a formula for a triangular decomposition of a
sum sequence (which is A081113 in the OEIS, but not in the partitioning
which is obvious from the formula there), and another one which isn't (perhaps
only in a first difference or partial sum format).
If anyone has seen these before, please contact the inventor of these numbers.
My attempts to receive further information on how these were generated, failed.


ad> Delivered-To: mathar
ad> Date: Tue, 30 Nov 2010 06:17:12 -0800 (PST)
ad> From: adesola dauda  adesoladaudaATyahooDOTcom
ad> Subject: ON RESEARCH ASSISTANCE
ad> 
ad> Good day Sir. I was the one that wrote some months(May/June) ago on Ph.D. Research Assistance on Semigroups of Sequences: 1, 4, 17, 68, 259,950, . . . , . 
ad> In your reply:
ad> 
ad> "The description is rather simple and based on geometry. You take a standard
ad> chess board with variable edge length. Take the king of the chess game
ad> (which can only move to one of the 8 adjacent squares), place it initially
ad> at one side, and count the different paths. So for n=2 we  have . . . " . 
ad> 
ad> I did as directed but I could not get the formular even the reference you made to sequence A081113 was followed but no results.
ad> 
ad> 
ad> Equally, I also sent  Idempotent sequence: 1, 3, 8, 21, 56, 149, . . . to you but what I found on Encyclopedia of Integer sequences is: 1, 3, 8, 21, 56, 150, . . . . What could be the problem as I have been working on this problems for years without edge way.
ad> 
ad> Sir, I am not only requesting you to send the formulars to me but also to please re-direct me on how to get the problems solved and equally get me materials(if available) on a similar problems on Combinatorics Mathematics.
ad> 

ad> Date: Wed, 29 Dec 2010 02:04:13 -0800 (PST)
ad> From: adesola dauda adesoladaudaATyahooDOTcom
ad> Subject: ELEMENTS GENERATED
ad> 
ad> 
ad> Dear Sir,
ad> Once again thanks for the assistance you have been rendering. The triangular array you sent to me is just a guide to the sequences: 1, 4, 17, 68, 259, 950, . . . and 1, 3, 8, 21, 56, 149, . . . . while the triangular array for the two sequences as it appears in my research work are:
ad> 
ad> n/k  1    2    3    4    5    6
ad> 1     1
ad> 2     2   2
ad> 3     3   12   2
ad> 4     4   42   20   2
ad> 5     5   120 108  24   2
ad> 6     6   310  448 156 28  2
ad> 
ad> and
ad> 
ad> n/k  1   2    3    4   5   6  
ad> 1   1
ad> 2   2   1
ad> 3   3   4     1
ad> 4   4   12   4   1
ad> 5   5   32  14  4   1
ad> 6   6   80  44 14  4  1


The standard decomposition of A081113 as sum_{k=1..n) k*(n-k+1)*A026300(.,.) is
+1  = 1.
+2 +2  = 4.
+3 +8 +6  = 17.
+4 +18 +30 +16  = 68.
+5 +32 +81 +96 +45  = 259.
+6 +50 +168 +300 +300 +126  = 950.
+7 +72 +300 +704 +1035 +912 +357  = 3387.
and therefore different from the first triangle shown above.