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%I #12 Jun 21 2017 18:53:25
%S 0,0,3,25,208,1928,20328,244536,3347328,51858432,902874240,
%T 17523066240,375931514880,8842225904640,226294152053760,
%U 6258916573056000,185978410684416000,5906514709831680000,199606607730561024000,7150186413112651776000,270578540735613960192000
%N Number of comparisons required to sort all permutations of [n] by MTF sort.
%C MTF sort is an (inefficient) sorting algorithm: the first element that is smaller than its predecessor is moved to front repeatedly until the sequence is sorted. Comparisons of adjacent elements always begin at the front and are continued until the last or the next element to be moved is found.
%H Alois P. Heinz, <a href="/A280970/b280970.txt">Table of n, a(n) for n = 0..404</a>
%H Project Euler, <a href="https://projecteuler.net/problem=523">Problem 523: First Sort I</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sorting_algorithm">Sorting algorithm</a>
%H <a href="/index/So#sorting">Index entries for sequences related to sorting</a>
%F a(n) = a(n-1)*n + (n-1)! * (2^n+(n-3)*n/2) for n>1, a(0) = a(1) = 0.
%F a(n) ~ (n-1)! * 2^(n+1). - _Vaclav Kotesovec_, Jan 12 2017
%p a:= proc(n) option remember;
%p `if`(n<2, 0, a(n-1)*n + (n-1)! * (2^n+(n-3)*n/2))
%p end:
%p seq(a(n), n=0..20);
%p # second Maple program:
%p a:= proc(n) option remember;
%p `if`(n<7, [0$2, 3, 25, 208, 1928, 20328][n+1],
%p ((4*n^2-23*n+2)*a(n-1)-(5*n^3-28*n^2-n+54)*a(n-2)
%p +(2*n-4)*(n^3-2*n^2-24*n+52)*a(n-3)
%p -(4*n-8)*(n-4)*(n-3)^2*a(n-4))/(n-6))
%p end:
%p seq(a(n), n=0..20);
%t Flatten[{0, Simplify[Table[n!*(n*(n-5)/4 - Pi*I - 1 - 2^(1+n)*LerchPhi[2, 1, 1+n]) , {n, 1, 20}]]}] (* _Vaclav Kotesovec_, Jan 12 2017 *)
%Y Cf. A279683.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jan 11 2017
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