[seqfan] Re: A019575 challenge

[Ron Hardin]

Is this "Empirical" or is there an obvious reason

n indistinguishable balls into n boxes
T(n,k) = ways to get exactly k balls in the most-filled box
table starts

1
1 2
1 6 3
1 18 12 4
1 50 50 20 5
1 140 195 90 30 6
1 392 735 392 147 42 7
1 1106 2716 1652 672 224 56 8
1 3138 9912 6804 2970 1080 324 72 9
1 8952 35850 27600 12825 4950 1650 450 90 10
1 25652 128865 110715 54450 22022 7865 2420 605 110 11
1 73788 461175 440374 228294 96030 36036 12012 3432 792 132 12
1 212940 1645215 1740024 948090 412698 160888 56784 17745 4732 1014 156 13
1 616226 5855941 6838832 3907995 1754116 705341 259896 86632 25480 6370 1274 182 
14
1 1787606 20810153 26762645 16011905 7391475 3050775 1162800 406980 128520 35700 
8400 1575 210 15

take the series a(n) = T(n,n-k) of the kth term from the right end of each line.

Empirical: The (k+1)th difference of T(n,n-k) equals k+1 for n > 2k.

(I'd guess the possibility of two equally filled max boxes ruins the counting)

see http://rhhardin.home.mindspring.com/current4.txt for the full b-file



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



----- Original Message ----
> From: Ron Hardin <rhhardin at att.net>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Thu, August 19, 2010 9:12:59 PM
> Subject: [seqfan] Re: A019575 challenge
> 
> Curiously the number of NON-distinguishable balls problem seems not to be in 
> OEIS
> (either that or I have a bug in computing it)
> 
> so far,  triangle  tabl
> 
> 1,1,2,1,6,3,1,18,12,4,1,50,50,20,5,1,140,195,90,30,6,1,392,735,392,147,
> 42,7,1,1106,2716,1652,672,224,56,8,1,3138,9912,6804,2970,1080,324,72,9,
> 1,8952,35850,27600,12825,4950,1650,450,90,10,1,25652,128865,110715
> T(n,k)=Number  of arrangements of n indistinguishable balls in n boxes with the 
>
> maximum  number of balls in any box equal to k
> Table starts
> 1
> 1 2
> 1 6 3
> 1  18 12 4
> 1 50 50 20 5
> 1 140 195 90 30 6
> Table of n, a(n) for  n=1..423
> 
> current and irregularly updated b-file at 
> http://rhhardin.home.mindspring.com/current4.txt
> 
> 
> rhhardin at mindspring.com
> rhhardin at att.net (either)
> 
> 
> 
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