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A302924 Central moments of a Fibonacci-geometric probability distribution.

%I A302924 #18 May 26 2018 22:32:30
%S A302924 1,0,22,210,4426,102330,2906362,95952570,3622138906,153816150810,
%T A302924 7257695358202,376693381614330,21328770664314586,1308295248437904090,
%U A302924 86423208789970618042,6116714829331037666490,461779664078480243085466,37040796099362864616022170
%N A302924 Central moments of a Fibonacci-geometric probability distribution.
%C A302924 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.
%C A302924 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.
%C A302924 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 + ....
%C A302924 The mean of this distribution is 6. (See A302922.)
%C A302924 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 this sequence.
%C A302924 For n >= 1, a(n) appears to be even. Dividing these terms by 2 gives sequence A302925.
%C A302924 The raw moments (i.e., the moments about zero) also appear to be integers. This is sequence A302922.
%C A302924 The raw moments also appear to be even for n >= 1. Dividing them by 2 gives sequence A302923.
%C A302924 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.
%C A302924 The cumulants also appear to be even for n >= 0. Dividing them by 2 gives sequence A302927.
%C A302924 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.
%H A302924 Albert Gordon Smith, <a href="/A302924/b302924.txt">Table of n, a(n) for n = 0..300</a>
%H A302924 Christopher Genovese, <a href="http://www.stat.cmu.edu/~genovese/class/iprob-S06/notes/double-heads.pdf">Double Heads</a>
%F A302924 In the following,
%F A302924 F(k) is the k-th Fibonacci number, as defined in the Comments.
%F A302924 phi=(1+sqrt(5))/2 is the golden ratio, and psi=(1-sqrt(5))/2.
%F A302924 LerchPhi(z,s,a) = Sum_{k>=0} z^k/(a+k)^s is the Lerch transcendant.
%F A302924 For n >= 0:
%F A302924 a(n) = Sum_{k>=1} (((k-6)^n)(F(k-1)/2^k));
%F A302924 a(n) = Sum_{k>=1} (((k-6)^n)(((phi^(k-1)-psi^(k-1))/sqrt(5))/2^k));
%F A302924 a(n) = (LerchPhi(phi/2,-n,-5)-LerchPhi(psi/2,-n,-5))/(2 sqrt(5));
%F A302924 a(n) = Sum_{k=0..n} (binomial(n,k)*A302922(k)*(-6)^(n-k)).
%e A302924 a(0)=1 is the 0th central moment of the distribution, which is the total probability.
%e A302924 a(1)=0 is the 1st central moment, or the "mean about the mean". It is zero by definition of central moments.
%e A302924 a(2)=22 is the 2nd central moment, known as the variance or the square of the standard deviation. It measures how far integers following the distribution are from the mean by averaging the squares of their differences from the mean.
%t A302924 Module[{max, r, g, moments},
%t A302924   max = 17;
%t A302924   r = Range[0, max];
%t A302924   g[x_] := x^2/(4 - 2 x - x^2);
%t A302924   moments = r! CoefficientList[Normal[Series[g[Exp[x]], {x, 0, max}]], x];
%t A302924   Table[Sum[Binomial[n, k] moments[[k + 1]] (-6)^(n - k), {k, 0, n}], {n, 0, max}]
%t A302924 ]
%Y A302924 Central half-moments: A302925.
%Y A302924 Raw moments: A302922.
%Y A302924 Raw half-moments: A302923.
%Y A302924 Cumulants: A302926.
%Y A302924 Half-cumulants: A302927.
%Y A302924 Cf. A000629, A000670, A052841, A091346.
%K A302924 nonn
%O A302924 0,3
%A A302924 _Albert Gordon Smith_, Apr 15 2018