[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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