A302923 Raw half-moments of a Fibonacci-geometric probability distribution.

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

Offset: 1

Author

Albert Gordon Smith, Apr 15 2018

Keywords

• nonn

Comments

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.

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 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 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 half-moments.

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.

Examples

			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.
		

Links

• Albert Gordon Smith, Table of n, a(n) for n = 1..300
• Christopher Genovese, Double Heads

Crossrefs

Raw moments: A302922.

Central moments: A302924.

Central half-moments: A302925.

Cumulants: A302926.

Half-cumulants: A302927.

Cf. A000629, A000670, A052841, A091346.

Programs

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
]

Formula

In the following,

F(k) is the k-th Fibonacci number, as defined in the Comments.

phi=(1+sqrt(5))/2 is the golden ratio, and psi=(1-sqrt(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/(1-z)).

For n>=1:

a(n) = (1/2)A302922(n);

a(n) = (1/2)Sum_{k>=1} ((k^n)(F(k-1)/2^k));

a(n) = (1/2)Sum_{k>=1} ((k^n)(((phi^(k-1)-psi^(k-1))/sqrt(5))/2^k));

a(n) = (1/2)(Li(-n,phi/2)/phi-Li(-n,psi/2)/psi)/sqrt(5).

E.g.f.: (1/2)g(e^x) where g(x) = x^2/(4-2x-x^2) is the g.f. for the probability distribution.