OFFSET
0,2
COMMENTS
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.
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. They form sequence A302926.
The cumulants appear to be even for n >= 0. Dividing them by 2 gives this sequence.
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 appear to be even. Dividing them by 2 gives sequence A302925.
LINKS
Albert Gordon Smith, Table of n, a(n) for n = 0..300
Christopher Genovese, Double Heads
FORMULA
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.
EXAMPLE
a(0)=0 is half the 0th cumulant of the distribution. The 0th cumulant is always zero.
a(1)=3 is half the 1st cumulant, which is half the mean.
a(2)=11 is half the 2nd cumulant, which is half the variance.
MATHEMATICA
Module[{max, r, g},
max = 17;
r = Range[0, max];
g[x_] := x^2/(4 - 2 x - x^2);
(1/2) r! CoefficientList[Normal[Series[Log[g[Exp[x]]], {x, 0, max}]], x]
]
PROG
(PARI) concat(0, Vec(serlaplace(log(exp(2*x)/(4-2*exp(x)-exp(2*x))))/2)) \\ Michel Marcus, Apr 17 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Albert Gordon Smith, Apr 15 2018
STATUS
approved