OFFSET
1,1
COMMENTS
If F(k) is the k-th Fibonacci number, where F(0)=0, F(1)=1, and F(n)=F(n-1)+F(n-2), then p(k)=F(k-1)/2^k is a normalized probability distribution on the positive integers.
For example, it is the probability that k coin tosses are required to get two heads in a row, or the probability that a random series of k bits has its first two consecutive 1's at the end.
The g.f. for this distribution is g(x) = x^2/(4-2x-x^2) = (1/4)x^2 + (1/8)x^3 + (1/8)x^4 + (3/32)x^5 + ....
The n-th moments about zero of this distribution, known as raw moments, are defined by a(n) = Sum_{k>=1} (k^n)*p(k). They appear to be integers and form A302922.
The e.g.f. for the raw moments is g(e^x) = 1 + 6x + 58x^2/2! + 822x^3/3! + ....
For n >= 1, the raw moments appear to be even. Dividing them by 2 gives this sequence of raw half-moments.
The central moments (i.e., the moments about the mean) also appear to be integers. They form sequence A302924.
The central moments also appear to be even for n >= 1. Dividing them by 2 gives sequence A302925.
The cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), also appear to be integers. They form sequence A302926.
The cumulants also appear to be even for n >= 0. Dividing them by 2 gives sequence A302927.
LINKS
Albert Gordon Smith, Table of n, a(n) for n = 1..300
Christopher Genovese, Double Heads
FORMULA
In the following,
F(k) is the k-th Fibonacci number, as defined in the Comments.
phi=(1+sqrt(5))/2 is the golden ratio, and psi=(1-sqrt(5))/2.
Li(s,z) is the polylogarithm of order s and argument z.
When s is a negative integer as it is here, Li(s,z) is a rational function of z: Li(-n,z) = (z(d/dz))^n(z/(1-z)).
For n>=1:
a(n) = (1/2)A302922(n);
a(n) = (1/2)Sum_{k>=1} ((k^n)(F(k-1)/2^k));
a(n) = (1/2)Sum_{k>=1} ((k^n)(((phi^(k-1)-psi^(k-1))/sqrt(5))/2^k));
a(n) = (1/2)(Li(-n,phi/2)/phi-Li(-n,psi/2)/psi)/sqrt(5).
E.g.f.: (1/2)g(e^x) where g(x) = x^2/(4-2x-x^2) is the g.f. for the probability distribution.
EXAMPLE
a(1)=3 is half the first raw moment of the distribution. It is half the arithmetic average of integers following the distribution.
a(2)=29 is half the second raw moment. It is half the arithmetic average of the squares of integers following the distribution.
MATHEMATICA
Module[{max, r, g, rawMoments},
max = 17;
r = Range[0, max];
g[x_] := x^2/(4 - 2 x - x^2);
rawMoments = r! CoefficientList[Normal[Series[g[Exp[x]], {x, 0, max}]], x];
Rest[rawMoments]/2
]
CROSSREFS
KEYWORD
nonn
AUTHOR
Albert Gordon Smith, Apr 15 2018
STATUS
approved