[seqfan] Two-sided Bell Numbers

[Ron Hardin]

/tmp/ejg
T(n,k)=Number of length n 0..k arrays with new values introduced in order from both ends

Table starts
.....1........1.........1.........1.........1.........1.........1.........1
.....1........1.........1.........1.........1.........1.........1.........1
.....2........2.........2.........2.........2.........2.........2.........2
.....4........4.........4.........4.........4.........4.........4.........4
.....8........9.........9.........9.........9.........9.........9.........9
....16.......23........23........23........23........23........23........23
....32.......64........65........65........65........65........65........65
....64......186.......199.......199.......199.......199.......199.......199
...128......551.......653.......654.......654.......654.......654.......654
...256.....1645......2275......2296......2296......2296......2296......2296
...512.....4926......8313......8568......8569......8569......8569......8569
..1024....14768.....31439.....33794.....33825.....33825.....33825.....33825
..2048....44293....121637....140039....140580....140581....140581....140581
..4096...132867....477307....605869....612890....612933....612933....612933
..8192...398588...1888721...2718531...2794159...2795181...2795182...2795182
.16384..1195750...7509799..12564289..13280627..13298407..13298464..13298464
.32768..3587235..29940861..59419764..65597882..65851100..65852872..65852873
.65536.10761689.119550419.285878342.335521900.338654554.338694406.338694479

Column 1 is A000079(n-2)
Column 2 is A164039(n-2)
Diagonal is A007476

Empirical for column k:
k=1: a(n)=2*a(n-1) for n>2
k=2: a(n)=5*a(n-1)-7*a(n-2)+3*a(n-3) for n>4
k=3: a(n)=10*a(n-1)-37*a(n-2)+64*a(n-3)-52*a(n-4)+16*a(n-5) for n>6
k=4: [order 7] for n>8
k=5: [order 9] for n>10
k=6: [order 11] for n>12
k=7: [order 13] for n>14

Some solutions for n=10 k=4  
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0..
..1....1....1....1....0....1....0....1....0....1....1....0....0....1....1....0..
..1....0....2....0....1....2....1....0....0....2....2....0....1....1....2....1..
..0....1....3....1....2....0....0....2....0....2....1....1....0....1....2....0..
..1....2....0....0....0....1....2....3....0....3....1....2....1....1....2....1..
..0....2....0....2....2....1....0....3....1....1....0....1....2....2....2....0..
..0....1....2....1....2....1....0....2....1....3....2....1....1....1....1....2..
..0....1....1....2....1....1....1....0....1....2....1....1....1....2....1....1..
..1....1....1....1....1....1....0....1....0....1....1....0....1....1....1....1..
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0..

Ordering only at one end, instead of both, gives A108934 (transposed), Bell numbers


 
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