# [seqfan] Re: A family of quadratic recurrences

Richard Mathar mathar at strw.leidenuniv.nl
Fri Oct 9 18:17:47 CEST 2009

```jol> From: Jaume Oliver i Lafont <joliverlafont at gmail.com>
jol> To: seqfan at list.seqfan.eu
jol> Subject: [seqfan] Re: A family of quadratic recurrences
jol> ...
jol>
jol> Three below the triangular diagonal
jol> dd(n,3) (n>=0)
jol> 1, 16693, 9180001, 413100001, 8605784251, 108618935041, 960948624385,
jol> 6514285680001, 35888934262501, 167421211254001, 681363156782881,
jol> 2473991485789825, 8154004643898751, 24728914828800001,
jol> 69765206224896001, 184727733874283521, 462456818974233865,
jol> 1101331124212230001, 2507950331037900001, 5485144545677412001,
jol> 11565626945922798211,
jol>
jol> Is there a similar polynomial for this sequence?
jol> ..

The recurrence (chopping the first 2 terms) is
a(n)= 17*a(n-1) -136*a(n-2) +680*a(n-3) -2380*a(n-4) +6188*a(n-5) -12376*a(n-6)
+19448*a(n-7) -24310*a(n-8) +24310*a(n-9) -19448*a(n-10) +12376*a(n-11)
-6188*a(n-12) +2380*a(n-13) -680*a(n-14) +136*a(n-15) -17*a(n-16) +a(n-17)

and the associated g.f.

g := -(1+16676*x+8896356*x^2+259309552*x^3+2820215510*x^4+25650504616*x^6+
12299525382*x^5-648980640*x^11+419031014*x^12+80567880*x^14+24979669664*x^7+
4603873*x^16+1095874768*x^10+33852*x^18-575484*x^17-209476682*x^13-23019376*x^
15+13005573378*x^8+1987877660*x^9)/(x-1)^17
=
-33852*x-2494894802400/(x-1)^14-2614181169600/(x-1)^13-846780706800/(x-1)
^11-531242712000/(x-1)^16-6072779790/(x-1)^8-54/(x-1)^4-1516084970400/(x-1)^15
-77544/(x-1)^5-381163320/(x-1)^7-1817176145400/(x-1)^12-1/(x-1)-261272670120/(
x-1)^10-10303740/(x-1)^6-51511677840/(x-1)^9-81729648000/(x-1)^17

from which one can get a 16th order polynomial...

1+81/2*n+1890*n^3+837/2*n^2+28969/64*n^11+13065/128*n^12+1063/64*n^13+59/32*n^
14+284353/32*n^5+178107/16*n^6+656267/64*n^7+80421/16*n^4+1/8*n^15+1816949/256
*n^8+238843/64*n^9+11953/8*n^10+1/256*n^16
= 1+ n*(n^3+7*n^2+14*n+4)*(n+1)^3*(n+3)^4*(n+2)^5/256, n>1 (!)

Richard Mathar

```