FW: another Motzkin triangle, no shortage of'm

[Meeussen Wouter (bkarnd)]

>  
>  [1]
>  [1, 1]
>  [1, 2, 2]
>  [1, 3, 4, 4]
>  [1, 5, 8, 9, 9]
>  [1, 8, 16, 20, 21, 21]
>  [1, 13, 32, 45, 50, 51, 51]
>  [1, 21, 64, 101, 120, 126, 127, 127]
>  [1, 34, 128, 227, 289, 315, 322, 323, 323]
>  [1, 55, 256, 510, 697, 792, 826, 834, 835, 835]
>  [1, 89, 512, 1146, 1682, 1998, 2135, 2178, 2187, 2188, 2188]
> 
> made from the GFs produced by
> 0	1	0	0	0 ...
> 1	0	1	0	0
> 1	1	0	1	0
> 1	1	1	0	1
> 1	1	1	1	0
> ...
> 
> in words: a binary matrix A with 1 just above the diagonal and everywhere below it.
> This matrix represents the substitution game
> 1->2, 2->1, 3->124, 4->1235, .. , k->123..(k-1)
> where we count the number of terms after n substitutions starting on "1".
> 
> Choosing the matrix to be of dimension w ( w=5 above)
> we get the GF(5) = -((-1 + x + x^2)/(1 - 2*x - x^2 + x^3))
> 
> and generally  GF(5)= {{1,0,0,0,0}}.(1/(Id - x A[5])) . {{1},{1},{1},{1},{1}}   or
> GF(w)={Prepend[0 Range[w-1],1]}.Inverse[IdentityMatrix[w] -x  A[w] ] . Table[{1},{w}] 
> 
> A  w-parametric GF(w)  would be nice, but I couldn't spot it.
> 
> The sequences a(w) produced by these GF's are
> 
> {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}
> {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597}
> {1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768}
> {1, 1, 2, 4, 9, 20, 45, 101, 227, 510, 1146, 2575, 5786, 13001, 29213, 65641, 147494}  (*A052534*)
> {1, 1, 2, 4, 9, 21, 50, 120, 289, 697, 1682, 4060, 9801, 23661, 57122, 137904, 332929} (*A018905, A024537*)
> {1, 1, 2, 4, 9, 21, 51, 126, 315, 792, 1998, 5049, 12771, 32319, 81810, 207117, 524394}
> {1, 1, 2, 4, 9, 21, 51, 127, 322, 826, 2135, 5545, 14445, 37701, 98514, 257608, 673933}
> {1, 1, 2, 4, 9, 21, 51, 127, 323, 834, 2178, 5734, 15183, 40365, 107614, 287457, 768875}
> {1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2187, 5787, 15435, 41419, 111659, 302059, 819243}
> {1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5797, 15499, 41746, 113116, 307976, 841723}
> {1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15510, 41822, 113531, 309937, 850118}
> 
> and these pass through some 'familiar' stages.
> Now  read columnwise from first row down to the diagonal,
> and you get the triangle this mail started from.
> The row sums are unfamiliar:
> 0 1 3 8 23 66 192 560 1639 4806 14116 
> 
> maybe this can be of help to prove that A018905 = A024537.
> 
> doodlingly yours,
> 
> W.
> 
> (may all your eggs be Fabergé)
> 


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