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 cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), appear to be integers and form this sequence.
The cumulants appear to be even for n >= 0. Dividing them by 2 gives sequence
A302927.
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 also appear to be integers and form sequence
A302922.
For n >= 1, the raw moments also appear to be even. Dividing them by 2 gives sequence
A302923.
The central moments (i.e., the moments about the mean) also appear to be integers. They form sequence
A302924.
For n >= 1, the central moments also appear to be even. Dividing them by 2 gives sequence
A302925.
Note: Another probability distribution on the positive integers that has integral moments and cumulants is the geometric distribution p(k)=1/2^k. The sequences related to these moments are
A000629,
A000670,
A052841, and
A091346.
