# [seqfan] Re: Excess permutation left moves over right moves

Ron Hardin rhhardin at att.net
Sun Jul 18 15:51:30 CEST 2010

```For the fourth and probably last recurrences the current actual elements (off
the end of the %STU lines) are
A000014
1 7736
2 34659
3 138022
4 511412
5 1807840
6 6188910
7 20713252
8 68199003
9 221845712
10 715077749
11 2288733954
12 7285025030
13 23085311544
14 72888488904
15 229435695984
16 720341436677
17 2256533814344
18 7054866364855
19 22017726215966
20 68606973122472
21 213470692870960
22 663337457939858
23 2058739522224316
24 6382300556437311
25 19764947385635104
26 61148579605356969
27 189006160887318266
28 583698202687347482
29 1801130778840866376
30 5553495149437365068
31 17110783201177674888
32 52683223242719095689

A000022
1 13831
2 54048
3 197961
4 693696
5 2357108
6 7835766
7 25638762
8 82915640
9 265808957
10 846445116
11 2681461839
12 8459737368
13 26600930930
14 83414901730
15 260966701576
16 814822256976
17 2539718326051
18 7903832697048
19 24563299810981
20 76240724337360
21 236365336287840
22 732006758451342
23 2264715211494710
24 7000157025711240
25 21618362711489337
26 66708490575450740
27 205685167948102091
28 633733656206606184
29 1951233763554312126

if you need them for the g.f., both with recurrrence
Empirical fit from a(1..24), then matches a(25..32):
a(n)=20*a(n-1)-180*a(n-2)+964*a(n-3)-3422*a(n-4)+8484*a(n-5)-15068*a(n-6)+19324*a(n-7)-17769*a(n-8)+11432*a(n-9)-4888*a(n-10)+1248*a(n-11)-144*a(n-12)

Incidentally the http://rhhardin.home.mindspring.com/current3.txt A-numbers can
shift as new sequences
meet the minimum term requirement and insert themselves; though I think they're
probably stable now
except for adding more at the end.

rhhardin at mindspring.com
rhhardin at att.net (either)

----- Original Message ----
> From: Richard Mathar <mathar at strw.leidenuniv.nl>
> To: seqfan at seqfan.eu
> Sent: Sun, July 18, 2010 9:16:46 AM
> Subject: [seqfan] Re: Excess permutation left moves over right moves
>
>
> Observation on the recurrences devlopped in the URL quoted in
> http://list.seqfan.eu/pipermail/seqfan/2010-July/005313.html :
>
> There is a pattern in the factored denominators of the empirical  g.f.'s:
> A000011:
>  -x*(-31+89*x-83*x^2+26*x^3) / ( (2*x-1)*(x-1)^3  ).
>
> A000012:
>  -x*(-146+776*x-1620*x^2+1677*x^3-868*x^4+180*x^5) / (  (2*x-1)^2*(x-1)^4 ).
>
> A000013:
>
>-x*(1289-13187*x+58154*x^2-144731*x^3+222774*x^4-217511*x^5+131690*x^6-45220*x^7+6744*x^8)
>  / ( (-1+3*x)*(2*x-1)^3*(x-1)^5 ).
>
> A000019:
>   -x*(-37+112*x-110*x^2+36*x^3) / ( (2*x-1)*(x-1)^3 ).
>
> A000020:
>   -x*(-219+1218*x-2658*x^2+2866*x^3-1536*x^4+328*x^5) / ( (2*x-1)^2*(x-1)^4
).
>
> A000021:
>
>-x*(1823-19427*x+89014*x^2-229394*x^3+364170*x^4-365192*x^5+226176*x^6-79168*x^7+12000*x^8)
>  / ( (-1+3*x)*(2*x-1)^3*(x-1)^5 ).
>
> In all these cases a decomposition of  the g.f. into partial
> fractions leads to closed forms as polynomials in n  multiplied by 1^n, 2^n or
> 3^n, because the denominators split into factors  linear in x.  Example
>A000021:
> -500 +12/(x-1)^3 -26/(x-1)^2  -729/(-1+3*x) +160/(2*x-1) -16/(2*x-1)^3
>+45/(x-1) -3/(x-1)^4 -1/(x-1)^5
> is  --besides the constant which doesn't matter with offset 1-- the sum  of
> 12/(x-1)^3-26/(x-1)^2+45/(x-1)-3/(x-1)^4-1/(x-1)^5
>   representing  -181/24*n^2-569/12*n-85-1/12*n^3+1/24*n^4
> and
> -729/(-1+3*x)
>    representing 729*3^n
> and
> 160/(2*x-1)-16/(2*x-1)^3
>   representing  (-144+8*n^2+24*n)*2^n. Total:
> A000021(n) =
>-181/24*n^2-569/12*n-85-1/12*n^3+1/24*n^4+729*3^n+(-144+8*n^2+24*n)*2^n
>
> Richard  Mathar
>
>
> _______________________________________________
>
> Seqfan  Mailing list - http://list.seqfan.eu/
>

```