

A330860


Denominator of the rational number A(n) that appears in the formula for the nth cumulant k(n) = (1)^n*2^n*(A(n)  (n  1)!*zeta(n)) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, with A(0) = 1 and A(1) = 0.


9



1, 1, 4, 8, 144, 3456, 172800, 10368000, 3810240000, 177811200000, 9957427200000, 75278149632000000, 1912817782149120000000, 53023308921173606400000000, 17742659631203112173568000000000, 426249654980023566857797632000000000, 9600207854287580784554747166720000000000
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OFFSET

0,3


COMMENTS

Hennequin conjectured his cumulant formula in his 1989 paper and proved it in his 1991 thesis.
First he calculates the numbers (B(n): n >= 0), with B(0) = 1 and B(0) = 0, given for p >= 0 by the recurrence
Sum_{r=0..p} Stirling1(p+2, r+1)*B(pr)/(pr)! + Sum_{r=0..p} F(r)*F(pr) = 0, where F(r) = Sum_{i=0..r} Stirling1(r+1,i+1)*G(ri) and G(k) = Sum_{a=0..k} (1)^a*B(ka)/(a!*(ka)!*2^a).
Then A(n) = L_n(B(1),...,B(n)), where L_n(x_1,...,x_n) are the logarithmic polynomials of Bell.
Hoffman and Kuba (2019, 2020) gave an alternative proof of Hennequin's cumulant formula and gave an alternative calculation for the constants (2)^n*A(n), which they denote by a_n. See also Finch (2020).
The Maple program given in A330852 is based on Tan and Hadjicostas (1993), where the numbers (A(n): n >= 0) are also tabulated.
For a list of references about the theory of the limiting distribution of the number of comparisons in quicksort, which we do not discuss here, see the ones for sequence A330852.


REFERENCES

Pascal Hennequin, Analyse en moyenne d'algorithmes, tri rapide et arbres de recherche, Ph.D. Thesis, L'École Polytechnique Palaiseau (1991), p. 83.


LINKS

S. B. Ekhad and D. Zeilberger, A detailed analysis of quicksort running time, arXiv:1903.03708 [math.PR], 2019. [They have the first eight moments for the number of comparisons in quicksort from which Hennequin's first eight asymptotic cumulants can be derived.]


EXAMPLE

The first few fractions A(n) are
1, 0, 7/4, 19/8, 937/144, 85981/3456, 21096517/172800, 7527245453/10368000, 19281922400989/3810240000, 7183745930973701/177811200000, ...
The first few fractions (2)^n*A(n) (= a_n in Hoffman and Kuba and in Finch) are
1, 0, 7, 19, 937/9, 85981/108, 21096517/2700, 7527245453/81000, 19281922400989/14883750, 7183745930973701/347287500, ...


MAPLE

# The function A is defined in A330852.
# Produces the sequence of denominators of the A(n)'s.
seq(denom(A(n)), n = 0 .. 40);


CROSSREFS



KEYWORD

nonn,frac


AUTHOR



STATUS

approved



