The Bell system

[Russell Walsmith]

I was playing around with Maple 10, writing code that generates e = exp(1)
as the sum of j/j! for j = 1,2,3... After finding some different ways to do
that, it occurred to me to look at sum (j*(i^j)/j!) (where i is the
imaginary unit). Generalized to j^n*(i^j)/j!, n = 1,2,3... this gives a
string of complex numbers, c[n]. When I tried abs(c[n]^2), I got the
sequence 1,1,2,9,61,554,6565,96677... = A121870 in the OEIS. But the Re/Im
components of the c[n] weren't interesting, until I remembered that
sum(j^n/j!)/e gives the Bell numbers B[n] = 1,1,2,5,15... = A000110.

The sequence of sum(j^n*(i^j)/j!)/e wasn't interesting, but
(j^n*(i^j)/j!)/exp(i) was the proper ploy. With the i^(j+1) adjustment, I
got real(c[n]) = -1,-1,-1,0,5,23,74,161,-57,-3466,-27361... which, except
for the initial term, is A121868. We also find the terms of imag(c[n]) =
0,0,-1,-3,-6,-5,33,266,1309... in the database as A121867 = 1,0,-1,-3,-6,-5,
33,266,1309... Note that i^(j-1) reverses the signs of the sequences, giving
the alternative versions of A121868.and A121867, and i^(j) swaps the real
and imag parts of c[n]. Abs(c[n]^2/exp(i)) too is A121870 because
abs(exp(i)) = 1.

Let P = a[1]*j^n+a[2]*j^(n-1)... +a[n]*j. Then exp(-1)*sum(P/j!) gives an
integer sequence for any such polynomial: e.g., (sum(j^(n-1)+j^(n-2))/j!)/e
= 1,2,3,7,20,67,255,1080,5017,25287,137122... = A011968. I eventually
realized that the polynomial P = a[1]*j^n+a[2]*j^(n-1)... +a[n]*j is, in
this context, a number in which place/value corresponds to ...B[5], B[4],
B[3], B[2], B[1] =   ...52,15,5,2,1. So in this "Bell system", the number 1
= decimal 1, 10 = decimal 2, 100 = decimal 5 and so on. That is, A000110 in
its native terms is 1,1,10,100,1000... Hence, we can express A024716 =
1,3,8,23,75,278... in the Bell system as 1,11,111,1111... and A011968
(referenced above) as 1,10,11,110,1100,11000...

Assuming that this ground hasn't already been explored more thoroughly
elsewhere, perhaps it's worth investigating whether symmetric Bell system
patterns will yield interesting decimal system sequences. E.g,
1,21,321,4321,54321 looked promising at first, giving the first four terms
of A003947 correctly, and 1,12,123,1234,12345 likewise gives the first four
terms of A079736. Both diverge from the database sequences at n=5, but maybe
they are somehow interesting in their own right. Moreover, for P with
imaginary coefficients, (P/j!)/exp(i) trials have so far generated integral
a + bi, so these techniques evidently provide means for generating a
plethora of new sequences... the question now is which of them are
remarkable. Perhaps some those on this list will have insights into that...

Russell
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