improvisation on A006319 à la A001003 and A000108

[wouter meeussen]

ID Number :   A006319 (Formerly M3521) 
Sequence :    1, 4, 16, 68, 304, 1412, 6752, 33028, 164512, 831620,..
Name : Royal paths in a lattice (convolution of A006318).

also results from the lengths of strings made with:
Replace integer k with the sequence -Abs[k] .. Abs[k]+1, and repeat n times:
1
-1,0,1,2
-1, 0, 1, 2, 0, 1, -1, 0, 1, 2, -2, -1, 0, 1, 2, 3
...

or, as sub-lists:

{{1},
{{-1, 0, 1, 2}}, 
{{{-1, 0, 1, 2}, {0, 1}, {-1, 0, 1, 2}, {-2, -1, 0, 1, 2, 3}}}}
...

or, in Mathematica lingo:

Table[Length@
    Flatten[Nest[(#/. k_Integer :> Range[-Abs[k],Abs[k]+1])&, {1}, n]],
{n,10}]

{4, 16, 68, 304, 1412, 6752, 33028, 164512, 831620, 4255728}

Remark that the sum of the sublists equals its last element,
and that equals 1+ Abs[the previous iterate], 

Also remark that the sum of terms equals half the length.
it= {2, 8, 34, 152, 706, 3376, 16514, 82256, 415810, 2127864}


Superseeker found that this last one (="it") fits the inverse binomial transform of
A071356= 1,2,6,20,72,272,1064,4272,17504,72896,307648,1312896,5655808,24562176,

Table[Sum[(-1)^(n-k) Binomial[n,k]( it[[k+1]] ), {k,0,n}], {n,9}]
{6, 20, 72, 272, 1064, 4272, 17504, 72896, 307648}


I wish I fully understood the ramifications,

Wouter.
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