more 'observational math'

[wouter meeussen]

dear seqfans,

what follows is a resumé. I think I exhausted the easy cases.
Will not push this further. (dead end?).

************************
Counting lengths of nested lists generates familiar seq's,
as
A000108 from k -> 0,..,k+1 ; start {0} (* only widely known one *)
A001003 from k-> 0,..,k+1,..,0 ; start {0}
A006319 from k -> -Abs[k],..,Abs[k]+1 ; start {0};
    with again a link to (inverse binomial transform of A071356).
A062992 from k-> -1 - Abs[k],.., Abs[k] + 1 ; start {1}
A052709 from k ->  -Abs[k + 1],.., Abs[-k + 1] start {0}
    but start at {1} gives an ugly cubic revogf
 A027307 from k->  -Abs[k + 1],.., Abs[-k + 1] step 2, start {1}; and take
odd indexed terms.
    start {0} : half previous with prepend 1.

************************

Wouter.



PS_1. the ugly cubic
[n+(-1-3n) a[n] + (-2 n-2n^2) a[n]^2 - 2n^2 a[n]^3 ,revogf]
gives:
1,3,11,41,159,633,2575,10657,44735,190017,815231,3527681,
15378687,67478401,297777407,1320753665,5884652543,26326301697,
118211192831,532574203905,2406726828031,10906541371393

PS_2 : sequence whose odd indexed terms are A034015=A027307/2 :
Rest @ CoefficientList[InverseSeries[Series[
(-1-6n-8n^2+(1+2n)^2Sqrt[1+4n])/(2(n+4n^2+4n^3)),{n,0,28}]],n]

{1,2,5,12,33,86,249,680,2033,5722,17485,50260,156033,455534,
1431281,4228752,13412193,40003058,127840085,384232156,1235575201,
3737280582,12080678505,36736735672,119276490193,364372758986,
1187542872989,3642094268836}