# [seqfan] Re: A family of quadratic recurrences

Jaume Oliver i Lafont joliverlafont at gmail.com
Fri Oct 9 16:30:47 CEST 2009

```Dear seqfans,

If we define the matrix
a(n,l)=if(n<l,1,sum(i=1,l-1,a(n-i,l)*sum(j=i,l-1,a(n-j,l)))/a(n-l,l))
and the element "n" in diagonal "d" as dd(n,d)=a(n+2+d,n+2),
the first three non-unit diagonals are fitted by similar polynomials.

On the main diagonal, with triangular numbers only:
dd(n,0) (n>=0)
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66,
d(n)=n*(n+3)/2+1

One below the triangular diagonal
dd(n,1) (n>=0)
1, 13, 51, 136, 295, 561, 973, 1576, 2421, 3565, 5071,
d(n)=n*(n+2)*(n+3)^2/4+1

Two below the triangular diagonal
dd(n,2) (n>=0)
1, 217, 3001, 20251, 92611, 329281, 979777, 2551501, 5989501, 12937321,
d(n)=n*(n+1)*(n+2)^3*(n+3)^3/16+1

Three below the triangular diagonal
dd(n,3) (n>=0)
1, 16693, 9180001, 413100001, 8605784251, 108618935041, 960948624385,
6514285680001, 35888934262501, 167421211254001, 681363156782881,
2473991485789825, 8154004643898751, 24728914828800001,
69765206224896001, 184727733874283521, 462456818974233865,
1101331124212230001, 2507950331037900001, 5485144545677412001,
11565626945922798211,

Is there a similar polynomial for this sequence?

Jaume

```