[seqfan] The mean and variance of a sequence (new transforms)

Neil Sloane njasloane at gmail.com
Tue Mar 20 21:56:55 CET 2018


Dear Seq Fans,
Given an infinite sequence s(m) (I assume m >= 1 for simplicity, but that's
not important, it is trivial to change the formulas) of integers, or
rationals, we can define the associated sequences of means and variances as
follows:

m(n) = (1/n) * Sum_{i=1..n} s(i), n>= 1,

v(1) = 0, and for n >= 2,
v(n) = (1/(n-1)) * Sum_{i=1..n} (s(i)-m(n))^2.

Of course m(n) and v(n) are rationals, so we get 6 sequences,
numerators of m(n), denominators of m(n),
ditto for variances,
nearest integer to m(n), ditto for v(n).

There's a belief in certain circles that
if the sequence is nonnegative then m(n) and v(n) are usually related by
something
called Taylor's Law, which says that for large n, v(n) approx-equals
A*m(n)^B
for constants A and B.  This paper

Joel E. Cohen, Statistics of Primes (and Probably Twin Primes) Satisfy
Taylor’s Law from Ecology, The American Statistician, 70 (2016), 399-404.

shows that this is true for the primes, and this led me to create the
following sequences:

Mean and variance of primes: A301273/A301274, A301275/A301276, A301277,
A273462.
(The last one was already there)

A sequel to that paper recently appeared, or is about to appear:

Simon Demers, Taylor's Law Holds for Finite OEIS Integer Sequences and
Binomial Coefficients, American Statistician, online: 19 Jan 2018;
https://doi.org/10.1080/00031305.2017.1422439;
https://www.tandfonline.com/doi/abs/10.1080/00031305.2017.1422439

where he checked that Taylor's Law holds for 100 "nice" sequences in the
OEIS.
("Finite" in his title refers to the fact that he looked at the first n
terms. The sequences he looked at were all infinite).  He doesn't say much
about the details.

I looked at the mean and variance of the terms in the n-th row of Pascal's
triangle,
and this produced these sequences:

Mean and variance of n-th row of Pascal's triangle: A084623/A000265,
A301278/A301279, A054650, A301280.

One could now do the same thing for any of the core sequences, and this
will probably produce a lot of new sequences (or link together old
sequences).
I invite you all to join the fun and send in these sequences

What about Catalans, Motzkins, squares, powers of 2, 3, etc, Fibonaccis,
Bernoullis,
harmonic numbers s(n) = Sum_{i=1..n} 1/i, etc ?
For Bernoullis we should do two versions, one for all B_n, and one for
B_{2n}

I can send copies of the above papers to anyone interested.


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