A302926 Cumulants of a Fibonacci-geometric probability distribution.

0, 6, 22, 210, 2974, 56130, 1324222, 37489410, 1238235454, 46740118530, 1984855550782, 93653819396610, 4860878501987134, 275227990564092930, 16882335978752910142, 1115211301788480951810, 78930528072274523870014, 5958837996496319756259330

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 cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), appear to be integers and form this sequence.

The cumulants appear to be even for n >= 0. Dividing them by 2 gives sequence 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.

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 also appear to be even. Dividing them by 2 gives sequence A302925.

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.

Variant of A103437. - R. J. Mathar, Jun 09 2018

Examples

			a(0)=0 is the 0th cumulant of the distribution. The 0th cumulant is always zero.
a(1)=6 is the 1st cumulant, which is always the mean.
a(2)=22 is the 2nd cumulant, which is always the variance.
		

Links

• Albert Gordon Smith, Table of n, a(n) for n = 0..300
• Christopher Genovese, Double Heads

Crossrefs

Half-cumulants: A302927.

Raw moments: A302922.

Raw half-moments: A302923.

Central moments: A302924.

Central half-moments: A302925.

Cf. A000629, A000670, A052841, A091346, A103437.

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[Log[g[Exp[x]]], {x, 0, max}]], x]
]

Formula

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