[seqfan] strangeness in A008290 : Triangle T(n, k) of rencontres numbers

Wouter Meeussen wouter.meeussen at telenet.be
Sun Sep 8 19:32:56 CEST 2013


call for help/explanation/insight :   sum(k=1 ..n;  k^w  A008290(n,k) ) = 
B(w) n!  for n>=w

with B(w) = 1,2,5,15,52,203,877,4140,21147,115975,578570,... (A000110 : Bell 
or exponential numbers)
why should this be?

For those who wonder what happens for n<w, look at
sum(k=1 ..n;  k^w/n!  A008290(n,k) ) for different values of n (within rows) 
and k (successive rows)


0    1    1    1    1    1    1    1    1    1    1    1    1
0    1    2    2    2    2    2    2    2    2    2    2    2
0    1    4    5    5    5    5    5    5    5    5    5    5
0    1    8    14    15    15    15    15    15    15    15    15    15
0    1    16    41    51    52    52    52    52    52    52    52    52
0    1    32    122    187    202    203    203    203    203    203    203 
203
0    1    64    365    715    855    876    877    877    877    877    877 
877
0    1    128    1094    2795    3845    4111    4139    4140    4140 
4140    4140    4140
0    1    256    3281    11051    18002    20648    21110    21146    21147 
21147    21147    21147
0    1    512    9842    43947    86472    109299    115179    115929 
115974    115975    115975    115975
0    1    1024    29525    175275    422005    601492    665479    677359 
678514    678569    678570    678570
0    1    2048    88574    700075    2079475    3403127    4030523 
4189550    4211825    4213530    4213596    4213597

If we dig a bit deeper, and look at the deficits of these rows versus their 
ultimate limit, we get a new surprise:

1    0    0    0    0    0    0    0
2    1    0    0    0    0    0    0
5    4    1    0    0    0    0    0
15    14    7    1    0    0    0    0
52    51    36    11    1    0    0    0
203    202    171    81    16    1    0    0
877    876    813    512    162    22    1    0
4140    4139    4012    3046    1345    295    29    1

with row sums = A005493 : a(n) = number of partitions of [n+1] with a 
distinguished block.
which calculates as Sum(k=1..n; k StirlingS2[n, k] ) for n=1 .. 16

It is A137650 : Triangle read by rows, A008277 * A000012.
calculated as
Sum(j=0..k; Binomial[k, j] BellB[n - k + j] ) for n=0..7 and k=0..n

So, a closed formula for sum(k=1 ..n;  k^w  A008290(n,k) ) is up for grabs. 
No?
Is this new?

Wouter.

--------------------------- for my record --------------------------
Table[Tr/@Table[k^w /n! 
Binomial[n,k]If[n-k==0,1,Round[(n-k)!/E]],{n,0,16},{k,0,n}],{w,16}];
Table[ BellB[w]-Tr/@Table[k^w /n! 
Binomial[n,k]If[n-k==0,1,Round[(n-k)!/E]],{n,0,15},{k,0,n}],{w,16}];
Table[ Sum[ StirlingS2[n,n-j],{j,0,n-k}],{n,7},{k, n}];







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