

A302923


Raw halfmoments of a Fibonaccigeometric probability distribution.


6



3, 29, 411, 7757, 183003, 5180909, 171119931, 6459325517, 274300290843, 12942639522989, 671756887456251, 38035572830424077, 2333081451314129883, 154118411443366428269, 10907930704590567517371, 823491157770358707135437, 66054810199299268861908123
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OFFSET

1,1


COMMENTS

If F(k) is the kth Fibonacci number, where F(0)=0, F(1)=1, and F(n)=F(n1)+F(n2), then p(k)=F(k1)/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/(42xx^2) = (1/4)x^2 + (1/8)x^3 + (1/8)x^4 + (3/32)x^5 + ....
The nth moments about zero of this distribution, known as raw moments, are defined by a(n) = Sum_{k>=1} (k^n)*p(k). They appear to be integers and form A302922.
The e.g.f. for the raw moments is g(e^x) = 1 + 6x + 58x^2/2! + 822x^3/3! + ....
For n >= 1, the raw moments appear to be even. Dividing them by 2 gives this sequence of raw halfmoments.
The central moments (i.e., the moments about the mean) also appear to be integers. They form sequence A302924.
The central moments also appear to be even for n >= 1. Dividing them by 2 gives sequence A302925.
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.
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, A091346.


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 kth Fibonacci number, as defined in the Comments.
phi=(1+sqrt(5))/2 is the golden ratio, and psi=(1sqrt(5))/2.
Li(s,z) is the polylogarithm of order s and argument z.
When s is a negative integer as it is here, Li(s,z) is a rational function of z: Li(n,z) = (z(d/dz))^n(z/(1z)).
For n>=1:
a(n) = (1/2)A302922(n);
a(n) = (1/2)Sum_{k>=1} ((k^n)(F(k1)/2^k));
a(n) = (1/2)Sum_{k>=1} ((k^n)(((phi^(k1)psi^(k1))/sqrt(5))/2^k));
a(n) = (1/2)(Li(n,phi/2)/phiLi(n,psi/2)/psi)/sqrt(5).
E.g.f.: (1/2)g(e^x) where g(x) = x^2/(42xx^2) is the g.f. for the probability distribution.


EXAMPLE

a(1)=3 is half the first raw moment of the distribution. It is half the arithmetic average of integers following the distribution.
a(2)=29 is half the second raw moment. It is half the arithmetic average of the squares of integers following the distribution.


MATHEMATICA

Module[{max, r, g, rawMoments},
max = 17;
r = Range[0, max];
g[x_] := x^2/(4  2 x  x^2);
rawMoments = r! CoefficientList[Normal[Series[g[Exp[x]], {x, 0, max}]], x];
Rest[rawMoments]/2
]


CROSSREFS

Raw moments: A302922.
Central moments: A302924.
Central halfmoments: A302925.
Cumulants: A302926.
Halfcumulants: A302927.
Cf. A000629, A000670, A052841, A091346.
Sequence in context: A262640 A302582 A335867 * A326433 A113871 A186451
Adjacent sequences: A302920 A302921 A302922 * A302924 A302925 A302926


KEYWORD

nonn


AUTHOR

Albert Gordon Smith, Apr 15 2018


STATUS

approved



