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 A302923 Raw half-moments of a Fibonacci-geometric probability distribution. 6
 3, 29, 411, 7757, 183003, 5180909, 171119931, 6459325517, 274300290843, 12942639522989, 671756887456251, 38035572830424077, 2333081451314129883, 154118411443366428269, 10907930704590567517371, 823491157770358707135437, 66054810199299268861908123 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. 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 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. 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 half-moments: A302925. Cumulants: A302926. Half-cumulants: 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

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Last modified September 28 14:14 EDT 2022. Contains 357070 sequences. (Running on oeis4.)