[seqfan] Symmetric Polynomials A209668 A209671 Turn up

[Ron Hardin]

http://oeis.org/search?q=id:A209668|id:A209671

turn up as lead recurrence coefficients in (in progress)


T(n,k)=Number of nXk 0..k-1 arrays with no column j greater than column j-1 in 
all rows

Table starts
.1....3.......10............35................126......................462
.1...15......568.........39695............4431876................724082352
.1...63....18226......14177855........23124921876...........68264066143602
.1..255...518320....4041974015.....85800824609376......4051316109991426752
.1.1023.14230810.1075113010175.285912852294921876.207406617181155352354002

Empirical for column k:
k=1: a(n)=a(n-1)
k=2: a(n)=5*a(n-1)-4*a(n-2)
k=3: a(n)=37*a(n-1)-279*a(n-2)+243*a(n-3)
k=4: a(n)=405*a(n-1)-43860*a(n-2)+1524160*a(n-3)-15636480*a(n-4)+14155776*a(n-5)

The coefficient of a(n-1) is A209671(k) (through at least k=1..7)

and

T(n,k)=Number of nXk 0..k-1 arrays with no column j greater than or equal to 
than column j-1 in all rows

Table starts
.1...1........1............1..................1........................1
.1...7......181........10311............1016501................152747323
.1..37.....9019......6470341........10058484751...........28744943858947
.1.175...331489...2509306671.....52311221188001......2438624218076957695
.1.781.10669771.801439905901.212180664326328751.153322267564381742818531

Empirical for column k:
k=1: a(n)=a(n-1)
k=2: a(n)=7*a(n-1)-12*a(n-2)
k=3: a(n)=55*a(n-1)-936*a(n-2)+4860*a(n-3)
k=4: 
a(n)=631*a(n-1)-144700*a(n-2)+15035200*a(n-3)-702208000*a(n-4)+11468800000*a(n-5)


The coefficient of a(n-1) is A209668(k) (through at least k=1..7)


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