OFFSET
1,2
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 mean of this distribution is 6. (See A302922.)
The n-th moments about the mean, known as central moments, are defined by a(n) = Sum_{k>=1} ((k-6)^n)p(k). They appear to be integers and form A302924.
For n >= 1, that sequence appears to be even. Dividing those terms by 2 gives this sequence.
The raw moments (i.e., the moments about zero) also appear to be integers. This is sequence A302922.
The raw moments also appear to be even for n >= 1. Dividing them by 2 gives sequence A302923.
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.
LerchPhi(z,s,a) = Sum_{k>=0} z^k/(a+k)^s is the Lerch transcendant.
For n>=1:
a(n) = (1/2)A302924(n);
a(n) = (1/2)Sum_{k>=1} (((k-6)^n)(F(k-1)/2^k));
a(n) = (1/2)Sum_{k>=1} (((k-6)^n)(((phi^(k-1)-psi^(k-1))/sqrt(5))/2^k));
a(n) = (1/2)(LerchPhi(phi/2,-n,-5)-LerchPhi(psi/2,-n,-5))/(2 sqrt(5));
a(n) = (1/2)Sum_{k=0..n} (binomial(n,k)A302922(k)(-6)^(n-k)).
EXAMPLE
a(1)=0 is half the 1st central moment of the distribution, or half the "mean about the mean". It is zero by definition of central moments.
a(2)=11 is half the 2nd central moment, or half the variance, or half the square of the standard deviation.
MATHEMATICA
Module[{max, r, g, moments, centralMoments},
max = 17;
r = Range[0, max];
g[x_] := x^2/(4 - 2 x - x^2);
moments = r! CoefficientList[Normal[Series[g[Exp[x]], {x, 0, max}]], x];
centralMoments = Table[Sum[Binomial[n, k] moments[[k + 1]] (-6)^(n - k), {k, 0, n}], {n, 0, max}];
Rest[centralMoments]/2
]
CROSSREFS
KEYWORD
nonn
AUTHOR
Albert Gordon Smith, Apr 15 2018
STATUS
approved