# [seqfan] Quest of a trinagular decomposition of A081113

Richard Mathar mathar at strw.leidenuniv.nl
Mon Jan 3 21:46:05 CET 2011

```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).
My attempts to receive further information on how these were generated, failed.

ad> Date: Tue, 30 Nov 2010 06:17:12 -0800 (PST)
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> "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> 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> 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> 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> Date: Wed, 29 Dec 2010 02:04:13 -0800 (PST)
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> n/k  1    2    3    4    5    6
ad> 4     4   42   20   2
ad> 5     5   120 108  24   2
ad> 6     6   310  448 156 28  2
ad> n/k  1   2    3    4   5   6
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.

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