A302926 - OEIS

%I #21 Jun 09 2018 08:04:49

%S 0,6,22,210,2974,56130,1324222,37489410,1238235454,46740118530,

%T 1984855550782,93653819396610,4860878501987134,275227990564092930,

%U 16882335978752910142,1115211301788480951810,78930528072274523870014,5958837996496319756259330

%N Cumulants of a Fibonacci-geometric probability distribution.

%C 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 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 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 The cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), appear to be integers and form this sequence.

%C The cumulants appear to be even for n >= 0. Dividing them by 2 gives sequence A302927.

%C 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.

%C For n >= 1, the raw moments also appear to be even. Dividing them by 2 gives sequence A302923.

%C The central moments (i.e., the moments about the mean) also appear to be integers. They form sequence A302924.

%C For n >= 1, the central moments also appear to be even. Dividing them by 2 gives sequence A302925.

%C 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.

%C Variant of A103437. - _R. J. Mathar_, Jun 09 2018

%H Albert Gordon Smith, <a href="/A302926/b302926.txt">Table of n, a(n) for n = 0..300</a>

%H Christopher Genovese, <a href="http://www.stat.cmu.edu/~genovese/class/iprob-S06/notes/double-heads.pdf">Double Heads</a>

%F E.g.f.: log(g(e^x)) where g(x) = x^2/(4-2x-x^2) is the g.f. for the probability distribution.

%e a(0)=0 is the 0th cumulant of the distribution. The 0th cumulant is always zero.

%e a(1)=6 is the 1st cumulant, which is always the mean.

%e a(2)=22 is the 2nd cumulant, which is always the variance.

%t Module[{max, r, g},

%t   max = 17;

%t   r = Range[0, max];

%t   g[x_] := x^2/(4 - 2 x - x^2);

%t   r! CoefficientList[Normal[Series[Log[g[Exp[x]]], {x, 0, max}]], x]

%t ]

%Y Half-cumulants: A302927.

%Y Raw moments: A302922.

%Y Raw half-moments: A302923.

%Y Central moments: A302924.

%Y Central half-moments: A302925.

%Y Cf. A000629, A000670, A052841, A091346, A103437.

%K nonn

%O 0,2

%A _Albert Gordon Smith_, Apr 15 2018