A302922 Raw moments of a Fibonacci-geometric probability distribution.

1, 6, 58, 822, 15514, 366006, 10361818, 342239862, 12918651034, 548600581686, 25885279045978, 1343513774912502, 76071145660848154, 4666162902628259766, 308236822886732856538, 21815861409181135034742, 1646982315540717414270874, 132109620398598537723816246

Offset: 0

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 this sequence.

The e.g.f. for the raw moments is g(e^x) = 1 + 6x + 58x^2/2! + 822x^3/3! + ....

For n >= 1, a(n) appears to be even.

Dividing these terms by 2 gives sequence A302923.

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, and A091346.

Examples

			a(0)=1 is the 0th raw moment of the distribution, which is the total probability.
a(1)=6 is the 1st raw moment, known as the mean of the distribution. It is the arithmetic average of integers following the distribution.
a(2)=58 is the 2nd raw moment. It is the arithmetic average of the squares of integers following the distribution.
		

Links

• Albert Gordon Smith, Table of n, a(n) for n = 0..300
• Ignas Gasparavičius, Andrius Grigutis, and Juozas Petkelis, Picturesque convolution-like recurrences and partial sums' generation, arXiv:2507.23619 [math.NT], 2025. See p. 23.
• Christopher Genovese, Double Heads

Crossrefs

Raw half-moments: A302923.

Central moments: A302924.

Central half-moments: A302925.

Cumulants: A302926.

Half-cumulants: A302927.

Cf. A000629, A000670, A052841, A091346.

Programs

Mathematica

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

PARI

Vec(serlaplace(exp(2*x)/(4-2*exp(x)-exp(2*x)))) \\ Michel Marcus, Apr 17 2018

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>=0:

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

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

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

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

a(n) ~ n! * (5 - sqrt(5)) / (10 * (log(sqrt(5) - 1))^(n+1)). - Vaclav Kotesovec, Apr 13 2022