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A302927 Half-cumulants of a Fibonacci-geometric probability distribution.

%I A302927 #26 May 26 2018 22:33:43
%S A302927 0,3,11,105,1487,28065,662111,18744705,619117727,23370059265,
%T A302927 992427775391,46826909698305,2430439250993567,137613995282046465,
%U A302927 8441167989376455071,557605650894240475905,39465264036137261935007,2979418998248159878129665
%N A302927 Half-cumulants of a Fibonacci-geometric probability distribution.
%C A302927 If F(k) is the k-th Fibonacci number A000045(k), then p(k)=F(k-1)/2^k is a normalized probability distribution on the positive integers.
%C A302927 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 A302927 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 A302927 The cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), appear to be integers. They form sequence A302926.
%C A302927 The cumulants appear to be even for n >= 0. Dividing them by 2 gives this sequence.
%C 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.
%C A302927 For n >= 1, the raw moments also appear to be even. Dividing them by 2 gives sequence A302923.
%C A302927 The central moments (i.e., the moments about the mean) also appear to be integers. They form sequence A302924.
%C A302927 For n >= 1, the central moments appear to be even. Dividing them by 2 gives sequence A302925.
%C A302927 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 A302927 Albert Gordon Smith, <a href="/A302927/b302927.txt">Table of n, a(n) for n = 0..300</a>
%H A302927 Christopher Genovese, <a href="http://www.stat.cmu.edu/~genovese/class/iprob-S06/notes/double-heads.pdf">Double Heads</a>
%F A302927 E.g.f.: (1/2)*log(g(e^x)) where g(x) = x^2/(4-2*x-x^2) is the g.f. for the probability distribution.
%e A302927 a(0)=0 is half the 0th cumulant of the distribution. The 0th cumulant is always zero.
%e A302927 a(1)=3 is half the 1st cumulant, which is half the mean.
%e A302927 a(2)=11 is half the 2nd cumulant, which is half the variance.
%t A302927 Module[{max, r, g},
%t A302927   max = 17;
%t A302927   r = Range[0, max];
%t A302927   g[x_] := x^2/(4 - 2 x - x^2);
%t A302927   (1/2) r! CoefficientList[Normal[Series[Log[g[Exp[x]]], {x, 0, max}]], x]
%t A302927 ]
%o A302927 (PARI) concat(0, Vec(serlaplace(log(exp(2*x)/(4-2*exp(x)-exp(2*x))))/2)) \\ _Michel Marcus_, Apr 17 2018
%Y A302927 Cumulants: A302926.
%Y A302927 Raw moments: A302922.
%Y A302927 Raw half-moments: A302923.
%Y A302927 Central moments: A302924.
%Y A302927 Central half-moments: A302925.
%Y A302927 Cf. A000045, A000629, A000670, A052841, A091346.
%K A302927 nonn
%O A302927 0,2
%A A302927 _Albert Gordon Smith_, Apr 15 2018